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Multivariate Limited Translation Estimators

Permanent Link: http://ufdc.ufl.edu/UFE0021388/00001

Material Information

Title: Multivariate Limited Translation Estimators
Physical Description: 1 online resource (129 p.)
Language: english
Creator: Papageorgiou, Georgios
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Bayes estimators are used quite often in the theory and practice of statistics. Their popularity is attributed to the more efficient inference that they often lead to compared to classical frequentist procedures. However, the Bayes estimators can result in high Bayes risk when the prior distribution is misspecified. Additionally, they can result in high frequentist risk when they are used for estimating parameters which depart widely from the assumed prior means. We developed multivariate limited translation Bayes estimators of the normal mean vector which serve as a compromise between the Bayes and the maximum likelihood estimators. We demonstrated the superiority of the limited translation estimators over the usual Bayes estimators under misspecified priors and often also in terms of the frequentist risks under the situation stated in the previous paragraph. We also extended the above results and developed multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes estimators and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Georgios Papageorgiou.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ghosh, Malay.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021388:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021388/00001

Material Information

Title: Multivariate Limited Translation Estimators
Physical Description: 1 online resource (129 p.)
Language: english
Creator: Papageorgiou, Georgios
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Bayes estimators are used quite often in the theory and practice of statistics. Their popularity is attributed to the more efficient inference that they often lead to compared to classical frequentist procedures. However, the Bayes estimators can result in high Bayes risk when the prior distribution is misspecified. Additionally, they can result in high frequentist risk when they are used for estimating parameters which depart widely from the assumed prior means. We developed multivariate limited translation Bayes estimators of the normal mean vector which serve as a compromise between the Bayes and the maximum likelihood estimators. We demonstrated the superiority of the limited translation estimators over the usual Bayes estimators under misspecified priors and often also in terms of the frequentist risks under the situation stated in the previous paragraph. We also extended the above results and developed multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes estimators and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Georgios Papageorgiou.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ghosh, Malay.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021388:00001


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33991224a9974c98b07316f33c1ee6061ef69358







MULTIVARIATE LIMITED TRANSLATION ESTIMATORS


By
GEORGIOS PAPAGEORGIOU



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































S2007 Georgios Papageorgiou



































To my parents and siblings










ACKENOWLED GMENTS

First, and foremost, I would like to thank my advisor, Dr. Malay Ghosh, for

sharing with me his knowledge, for the invaluable guidance he offered me and for his

encouragement. I would also like to thank my committee members, Dr. Ramon Littell,

Dr. Ronald Randles, Dr. Andrew Rosalsky and Dr. Myron C'I I.1. for their constructive

comments and support. Special thanks also go to the faculty members of the Department

of Statistics at the University of Florida as well as to Dr. Janet Forrester of the Tufts

University for providing me with data sets and for her collaborative work.

Special thanks are also due to my friends for their support and encouragement.

Lastly, and most importantly, I would like to thank my entire family for their love and

support .











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... . .. .. 4

LIST OF TABLES ......... ... . 7

LIST OF FIGURES ......... .. . 9

ABSTRACT ......... ..... . 10

CHAPTER

1 INTRODUCTION AND LITERATURE REVIEW .... .. .. 11

1.1 Introduction ......... . .. .. 11
1.2 Overview of Limited Translation Estimators .... ... .. 12
1.2.1 The Bayes Case ......... .. 12
1.2.2 The Empirical BEi-;- Case . ...... .. 15

2 MULTIVARIATE LIMITED TRANSLATION BAYES ESTIMATORS .. .. 23

2.1 Bai-; R Estimators ......... ... 23
2.2 Limited Translation Bai-; R Estimators ..... .. .. 25
2.3 Bai-;- Risk of the Limited Translation Bai-; R Estimators .. .. .. 26
2.4 Frequentist Risk of the Limited Translation Bai-; R Estimators .. .. .. 30
2.5 Robustness of Limited Translation Bai-; R Estimators .. .. 34

3 MULTIVARIATE LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS: THE CASE OF UNKNOWN PRIOR
MEAN.. .. ................ ... 41

3.1 Bai-; <, Empirical BEi-;- Estimators and Influence Functions .. .. .. .. 41
3.2 Limited Translation Empirical BEi-;- Estimators ... .. .. .. 44
3.3 Bai-;- Risk of the Limited Translation EB Estimators .. .. .. .. 45
3.4 Frequentist Risk of the Limited Translation EB Estimators .. .. .. .. 48
3.5 Robustness of Limited Translation EB Estimators .. .. .. 51

4 MULTIVARIATE EMPIRICAL BAYES AND LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS . .... .. 55

4.1 Introduction ......... . .. 55
4.2 Empirical BEi-;- Estimators . ...... ... .. 57
4.3 Limited Translation Empirical BEi-;- Estimators ... .. .. .. 60
4.4 Bai-;- Risk of the Limited Translation EB Estimators .. .. .. .. 61
4.5 Frequentist Risk of the Limited Translation EB Estimators .. .. .. .. 64
4.6 A Simulation Study . .... .. 65










5 MULTIVARIATE EMPIRICAL BAYES AND LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS: THE CASE OF ALL UNKNOWN
PARAMETERS. ........ ......


Development of Estimators.
Bai-;- Risk of the EB Estimators
Bai-;- Risk of the Limited Translation EB Estimators.
Frequentist Risk of the EB Estimators
Frequentist Risk of the Limited Translation EB Estimators
A Simulation Study.
Application


5.6
5.7


II


6 SUMMARY AND CONCLUSIONS ..............

APPENDIX

A PROOF OF THEOREM 2 SERIES ..............

A.1 Two General Results ..........
A.2 Proof of Theorem 2.4.1 ...........
A.3 Proof of Corollary 2.4.2 ......... ... .
A.4 Proof of Theorem 2.5.1 ...........
A.5 Proof of Corollary 2.5.2 ......... ... .

B PROOF OF THEOREM 3 SERIES . ..... .. .

B.1 Proof of Theorem 3.4.1 ......... .. .
B.2 Proof of Theorem 3.5.1 ...........

C PROOF OF THEOREM 4 SERIES ..............

C.1 Proof of Theorem 4.2.1 ......... .. .
C.2 Proof of Theorem 4.2.3 ......... .. .
C.3 Proof of Theorem 4.4.1 ......... .. .
C.4 Proof of Theorem 4.5.1 ......... .. .

D PROOF OF THEOREM 5 SERIES . ..... .. .

D.1 Proof of Theorem 5.2.1 ................
D.2 Proof of Theorem 5.3.1 ................
D.3 Proof of Theorem 5.4.1 ................
D.4 Proof of Theorem 5.5.1 ................

REFERENCES .......... .........

BIOGRAPHICAL SKETCH ...........


94
97
100
101
103

104

104
105


107
108
110
115

117










LIST OF TABLES


Table page

2-1 Mininiun values, k, of A and P(A > k) for p = :3 .... . .. :39

2-2 Mininiun values, k, of A and P(A > k) for p = 5 .... . .. :39

2-3 Mininiun values, k, of A and P(A > k) for p = 10 .... .. :39

4-1 Comparison of the risks of the estiniators under the contaminated model and
the fl sampling distribution, n = :30. . ...... .. 70

4-2 Comparison of the risks of the estiniators under the contaminated model and
the f2 saI1pling distribution, n = :30. . ...... .. 70

4-3 Comparison of the risks of the estiniators under the normal model and the fl
sampling distribution, n = :30. ......... ... 70

4-4 Comparison of the risks of the estiniators under the normal model and the f2
sampling distribution, n = :30. ......... ... 71

4-5 Comparison of the risk of the estiniators under the contaminated model and
the fl sampling distribution, n = 20. . ...... .. 7:3

4-6 Comparison of the risk of the estiniators under the contaminated model and
the f2 saI1pling distribution, n = 20. . ...... .. 7:3

4-7 Comparison of the risk of the estiniators under the normal model and the fl
sampling distribution, n = 20. ......... ... 7:3

4-8 Comparison of the risk of the estiniators under the normal model and the f2
sampling distribution, n = 20. ......... ... 74

5-1 Comparison of the risks of the estiniators under the contaminated model and
the fl sampling distribution, n = :30. . ...... .. .. 88

5-2 Comparison of the risks of the estiniators under the contaminated model and
the f2 saI1pling distribution, n = :30. . ...... .. .. 88

5-3 Comparison of the risks of the estiniators under the normal model and the fl
sampling distribution, n = :30. ......... ... .. 88

5-4 Comparison of the risks of the estiniators under the normal model and the f2
sampling distribution, n = :30. ......... ... .. 89

5-5 Comparison of the risk of the estiniators under the contaminated model and
the fl sampling distribution, n = 20. . ..... .. 89

5-6 Comparison of the risk of the estiniators under the contaminated model and
the f2 saI1pling distribution, n = 20. . ..... .. 89










5-7 Comparison of the risk of the estiniators under the normal model and the fl
samplingf distribution, n = 20. ......... ... .. 90

5-8 Comparison of the risk of the estiniators under the normal model and the f2
samplingf distribution, n = 20. ......... ... .. 90

5-9 1\LE, EB, LT Estimates ......... . .. 91










LIST OF FIGURES


Figure page

2-1 Plot of 1 s, as a function of c. ......... ... 38

2-2 Plot of risk as function of the non-centrality parameter, X. .. .. 38

2-3 Bai-; a risk under misspecified priors. . .... .. 40

3-1 Risks plotted against the the non-centrality parameter, As. .. .. .. 54

3-2 Bai-; a risks plotted against the assumed parameter g when the true parameter
is taken to be g* = 2. .. ... ... .. 54

4-1 Plot of 1 s, as a function of c, for p = 2, 10 and n = 10, 30. .. .. .. .. 68

4-2 The Os generated from (a) the contaminated model and (b) the normal model. .69

4-3 A sample of Os generated from (a) the contaminated model and (b) the normal
model. ............ .............. 72

5-1 Plot of RLS=E{1 pc(aW)}2 aS fulCtilOn of c, for p = 2, 5 and n = 10, 30. .. 87

5-2 Average intakes of vitamins A and B1. . .... .. 91

5-3 Estimated intakes of vitamins A and B1. ..... .. .. 92









Abstract of dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MULTIVARIATE LIMITED TRANSLATION ESTIMATORS

By

Georgios Papageorgiou

August 2007

Cl. ny~: Malay Ghosh
Major: Statistics

Bai-;- estimators are used quite often in the theory and practice of statistics. Their

popularity is attributed to the more efficient inference that they often lead to compared to

classical frequentist procedures. However, the Bai-;- estimators can result in high Bai-i

risk when the prior distribution is misspecified. Additionally, they can result in high

frequentist risk when they are used for estimating parameters which depart widely from

the assumed prior means.

We developed multivariate limited translation Bai-;- estimators of the normal mean

vector which serve as a compromise between the Bai-;- and the maximum likelihood

estimators. We demonstrated the superiority of the limited translation estimators over the

usual Bai-;- estimators under misspecified priors and often also in terms of the frequentist

risks under the situation stated in the previous paragraph.

We also extended the above results and developed multivariate limited translation

empirical BEi-;- estimators of the normal mean vector which serve as a compromise

between the empirical BEi-;- estimators and the maximum likelihood estimators. These

compromise estimators perform better than the regular empirical BE-- a estimators, in

a frequentist sense, when there is wide departure of an individual observation from the

grand average.










CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW

1.1 Introduction

B li-, -i Ia methods are used extensively these d 7i-< in the theory and practice of

statistics. One appealing feature of the B li-, -i Ia procedure is that the posterior inference,

based on an approximately elicited prior along with the likelihood, usually leads to more

efficient inference than any classical frequentist inferential procedure. This is intuitively

true since one is utilizing two sources of information rather than one as in classical

analysis. Such elicitation of priors has been possible in the presence of extensive historical

data. A very important application is in the medical area where constant updating of

information leads to successful prior elicitation. Also, people in Educational Testing

Service (ETS) have been using B li-, -i Ia methods regularly because they have in store a

vast amount of test scores from multiple tests that they administered. In particular, IQ

test scores are better calibrated when one uses a prior along with the sample data.

A well known characteristic of the B li-, -i Ia methods of estimation for the exponential

family of distributions is that the Bai-;- estimators, namely the posterior means, shrink

the maximum likelihood estimators towards the means of the prior distributions, with the

extreme values experiencing the most shift. Although the Ba-- a estimators do, on average,

better than any other estimator, they may perform badly in individual cases where true,

unobservable means are far away from the assumed prior means. In addition, B li-, -i Ia

methods can lead to severely wrong conclusions if the assumed prior is widely different

from the 'true' prior.

Robust B li-, -i Ia methods have been proposed to guard against problems of this

type. One such procedure, first introduced by Efron & Morris (1971, 1972a, 1975), and

referred to as 'limited translation rules' by these authors, is the subject matter of this

dissertation. The limited translation rules are compromises between the Bai-;- and the

maximum likelihood estimators that slightly increase the Bai-; a risk but guard against










large frequentist risks and large Bayes risks for cases where the prior distributions are not

correctly specified.

One of the virtues of the limited translation rules is that they do not fare too badly

in their Bai-; a risk performance, compared to the regular Bayes estimators, even if the

assumed prior is true. On the other hand, if the assumed prior departs widely from

the true prior, then the limited translation rules do have much superior Box-; a risk

performance than the regular Bai-;- estimators. In a frequentist risk sense, the limited

translation rules do not perform too badly relative to the regular Bai-; a rules if the

parameter to be estimated is close to the assumed prior mean. On the other hand, if the

parameter to be estimated is far from the prior mean, then the limited translation rules do

perform much better than the regular Ba-- a estimators.

Efron & Morris (1971, 1972a, 1975) developed limited translation rules for the

univariate normal case. The objective here is to develop limited translation rules for

multivariate normal case. In this chapter we review the literature related to limited

translation estimators.

1.2 Overview of Limited Translation Estimators

1.2.1 The Bayes Case

Let X|8 ~ NV(0, 1) and 8 ~ ( NV(0, A). The interest here is in estimating the

unobservable quantity 8 based on an observation x on the variable X. Under the squared

error loss function, L(0, a) = (8 a)2, Or any Other increasing function of |8 a|, the Bai-i

estimator of 8 given X is


8"(X) =X =(1 B)Xrv, (1 1)
A+1

where B = (A + 1)-1

The estimating rule in Equation 1-1 is optimal in the sense that it minimizes the

expected risk, r((, 0) = E{0 0(X)}2, among all choices of estimating rules 0(X), with









the expectation being taken with respect to the joint distribution of X and 8 under the

assumed prior (.

The estimator 8" shrinks the maximum likelihood estimator (jll~l) of 8 towards the

mean of the prior distribution, p = 0, by a fixed proportion, 1 B. This type of shrinkage

estimators is reasonable if the assumed prior is close to the 'true' one. However, shrinkage

estimators perform poorly if the assumed and the 'true' prior are far apart. Suppose for

example that the 'true' prior distribution is 8 ~ (* NV(p*, A*). In this case, the Bai-i

risk of the estimator in Equation 1-1 is


r(I*, 0") = B2(11) + 2 2(A2 + A*). (1-2)


Clearly, if the true prior mean, p* is far from the assumed prior mean, p = 0, then the

Bai-; a risk of 0" is quite high.

Also, the frequentist risk, that is the risk as a function of 8, denoted by R(0, 0") and

calculated as


R(0, 0") = Ea(8 8 )2 = 282 + 1_ B 2, _q


will be high when 8 is far from the assumed prior mean, p~ = 0, the reason being that OE

shrinks the maximum likelihood estimator of 8, 8o = X, towards the prior mean p.

On the other hand, 8o = X has minimax risk equal to R(0, 8o) = 1 for all 8, and

thus the average risk of Bo, the average taken with respect to prior (, is r((, 8o) = 1 which,

however, is bigger than the average risk of the Bai-;- estimator, r((, 8 ) = 1 B.

In order to combine the good properties of the Bai-; a rule with those of the MLE,

that is to maintain low Bayes risk and at the same time put a bound to the frequentist

risk, Efron & Morris (1971, 1972a, 1975) proposed limited translation rules. These rules

are simple compromises between the Bai-; a rules and the MLEs.

For the estimation problem at hand, they proposed a rule akin to the Ba-, a rule,

which however does not allow deviations from 8o = X bigger than a fixed value, = a- m.









That is the rule satisfies the constraint that |0" Xl | m which holds true if |XI | d,

where d = mB-l. The limited translation rule of maximum translation m is defined as

X + m if X < -d

eTf = ( 1 B X if -d < X < d (1-4)

X m if X > d.

The limited translation rule can also be written as

mB- z mB-1 z
HL = {1 -min(1, ,)}HOomin(1, ,)8B

= 1 pc(BX2 )0 c p(BX2 8B

= 1 pc(BX2)B}X, (1-5)

where c = mB-5 and pc(u) = min(1, c/ ~).

The function pc(.) is termed relevance function. It measures the relevance of the

Bai-; a rule, and thus the relevance of the prior parameters, to the estimation of 8, while

1 pc(.) measures the relevance of the MLE. Its argument, BX2, iS the square of the

distance, measured in standard deviations, of the variable X from its mean value 0, under

the marginal distribution of X ~ NV(0, B-l).

The properties of the limited translation rule are examined by calculating its Bai-i

and frequentist risks. Efron & Morris (1971) showed that


r (i', HL) = T(E, 8o) (1 se) + T (l, 8 ) sc, (1 6)

where 1 s, is a decreasing convex function of c. It is calculated as 1 s, = E {1 pc(U)}2

where U ~ Xg. Also, these authors showed that


sup R(0, 8L) = 1 + m2 (7


The Bai-; a risk of the limited translation rule is a weighted average of the Ba-, a risks of

the Bai-; a rule and the MLE, the weights being s, and 1 s, respectively. This, of course,










means that the Bayes risk of the limited translation rule is higher than that of the Bai-i

rule. However, the weight of the risk of the MLE, 1 s,, is a decreasing convex function

of c. This allows the statistician to choose c by deciding by what proportion it is worth

increasing the Bai-; a risk of the Bai-; a rule in order to receive protection against large

frequentist risks. This protection becomes evident by comparing the frequentist risk of

the Bai-; a rule to the frequentist risk of the limited translation rule. The former is given

in Equation 1-3 and it is clearly an unbounded function of 8, while the supremum with

respect to 8 of the latter is 1 + m2

1.2.2 The Empirical Bayes Case

We now consider the scenario where the prior variance A is unknown and thus the

Bai-; a rule cannot be used as such. In this case however, A can be estimated from the

data and an unbiased estimator of B = (1 + A)-l can be inserted in Equation 1-1, thus

resulting in an empirical BEi-;- (EB) estimator. The EB estimators have become very

popular since Efron & Morris (1972a, 1973, 1975) gave the EB interpretation of the

celebrated James-Stein estimator, James & Stein (1961). Statisticians have applied these

methods to many important problems, in particular for simultaneous estimation of several

parameters.

Consider the case where we have p > 3 independent univariate normal observations


X | if (s 02,i= ,2 ..,p (1-8)


where O.2 = var(Xi) = 1. Letting X = (X1,...,X,)T and 8 = (01,...,0,)T, the sampling

distributions in 1-8 can more briefly written as Xl |9 ~N,(0, I,), where I, denotes the

identity matrix of order p.

Further, suppose that


Os i N~, A) i 1, .. p,(1-9)









where the prior mean, for the sake of simplicity, is assumed to be equal to 0, while the

prior variance, var(04) = A, is unknown. Equivalently, the prior distributions in 1-9 can be

written as 8 ~ ( NV,(0, AI,).

The goal here is to estimate the unknown Os, i = 1, 2, .. ,p. There are two competing

loss functions that we take into consideration. First, for the estimation of an individual

parameter 8i, we consider the individual squared error loss function


L(0s, ai) = (8i ai)2, _10)

where ai is the estimate of the true parameter Os. We also consider the ensemble, or total,

loss function given by adding the individual losses


L(0, a) = L(0s, a) = (4n)' as)2
i= 1 i= 1

where a = (al,..., a,)T is a vector guess for 8 = (8 ,...,0,) .

Under the distributional assumptions stated above, the posterior mean of Os given X ,

that is the Bai-;- estimator of 8i, is


Of = (1 B)Xi. (1-12)

We may recall that B = (1 + A)-l

Since A is unknown so is B, and the Bai-; a rule cannot be used as such. However,

B can be estimated from the marginal distribution of the data. Marginally X~

NV,(0, B-1I,), and thus B||X||2 X It follows that E(p 2/||X||2) = B. Thus,

B = (p 2)/||X||2 1S an unbiased estimator for B. Substituting this expression for B in

Equation 1-12 results in the celebrated James-Stein estimator for 8 namely


Of ( )X (1 1 17)(1BX,. (1-13)


In victor notations, EBC EB 10B TZ rtenR B _^)X.









Under the ensemble, or total, loss function of Equation 1-11, the risk of the MLE, X,

for all 8, is


R(0, X) = Eg {L(0,X)} = Easo(0i X,)2= 14
i= 1

Under the same loss function, James & Stein (1961) showed the risk of the James-Stein
estimator, OEB is


h'(. "R= E~l(B d~~R J~~lp- p p2 + 2K

where K ~ Poisson(A = 2-1||8||2. The term curly brackets in the right hand side of

Equation 1-15 is a strictly increasing concave function of A. It takes on its minimum

value of 2/p~ at A = 0 and it approaches its supremum of 1 as A increases. Thus, OEB

has uniformly lower risk than the MLE for every value of 8 in the sense of the total loss

function L(0, a).

Baranchik (1964) considered the risk that occurs when estimating an individual 04 by
the James-Stein estimator. He showed that the risk of 6,EB is


R(Bi, HO;f ) =Eg(04 Of")2

1 +2(p ||18||2 (p 2 + 2K)(p + 2K)
(p- 2)Eh E + (1 16)
(p 2 +2K) (p +2K)

where K ~ Poisson(A = 2-1||8||2). The risk in Equation 1-16 is maximized for fixed ||9||2

at 0,? = ||9||2, that is when the vector parameter 8 has all its components except the ith

equal to zero, giving the expression for the maximum risk as

I+(p 2)~h Ep 2 (1-17)
(p 2 +2K) (p +2K)

The expression in 1-17 attains its maximum of approximately p/4 near A = p/2.

The point of the above is that although the James-Stein estimator has smaller total

risk than the MLE, as we saw in Equation 1-15, it may do poorly in estimating individual










8i that have unusually small or large values. On the other hand, the MLE of 8i has

minimax risk equal to 1.

It is worth noting that the Bai-; a risk of the James-Stein rule, that is the total

squared error loss of Equation 1-11 averaged over the joint distribution of X and 8, is

given by r((, OEB) = p(1 B) + 2B. The first term in this expression, p(1 B), is the

risk of the Bai-; a rule. We can thus think of the second term as the the price for having to

estimate the prior variance, A, from the data. The Bai-; a risk of the James-Stein rule can

equivalently be written as r((, OEB) = p{1 B(p 2)/p}, where the term in curly brackets

is clearly less than one and thus the Bai-; a risk of OEB is leSS than p, the Bai-; a risk of the

MLE.

Efron & Morris (1972a) developed limited translation empirical BE-- a rules, a

compromise between the James-Stein rule and the MLE, in an effort to lower the

maximum risk of the individual components of the James-Stein rule and at the same

time maintain low total risk and low Bai-; a risk. The limited translation rule follows as

closely as possible the James-Stein rule provided that it does not deviate from the MLE by

more than a fixed value.

The EB, or James-Stein, estimator was obtained as the estimator of the Ba-, a rule,

by replacing (p 2)/||X||2 for B. Similarly, the limited translation EB rule is obtained

as the estimator of the limited translation Bai-; a rule. Recall that in Equation 1-5, the

limited translation Bai-; a rule was briefly written as 8 = {1 pc(BX2)B}X, where

pc(u) = min(1, c/~). Replacing B by its unbiased estimator results in the limited

translation EB estimator for 8i

OLEB _' -c X,. )~ (1-18)
I ||X||2 ||IX||2

In1 vector notation~s, OLEB =(~~! LE pLE HiT

Since the argument of the function pc is alv-ns-s less than or equal to p 2, the

values of c that need to be considered are those between 0 and zO. For c = 0 the










resulting estimator is the MLE, while for c = z0- the estimator is equivalent to

the James-Stein estimator. The value of c is chosen in such a way that the estimator

has the good individual properties of the MLE and the good ensemble properties of the

James-Stein estimator.

The relevance function, pc(u) = min(1, c/ ~), measures the relevance of the whole

to the individual. For instance, when XJ/||X||2 takes on a large value, that is when Xi

is much larger than the rest of the observations in absolute value, then the whole is not

considered to be very relevant for the estimation of the corresponding Os. In such a case,

the MLE is considered to be more relevant. The relevance function, by decreasing on its
domain, O < < p 2, allmow the estimator (LEB to deviate from the MLE by less as

X2/||X|| 2 inCTORS6S.

The properties of the limited translation EB rule are examined by first calculating its

Bai-; a risk. The result here is similar to the result of the Bai-;- case. The Bai-; a risk of the

limited translation EB rule can be shown to be equal to a weighted average of the Bai-i

risks of the MLE and the James-Stein estimator


r (I, OLEB) = r (, X) (1 se) + T (C, tjEB) Sc. (1-19)


The weight of the Bai-; a risk of the MLE, 1 s,, is calculated as 1 s, = E[1 p{((p -

2)W,}]2, Where W, ~ Beta(3/2, (p- 1)/2), and for fixed p it is a decreasing convex function

of c. Therefore the inverse function is also defined c(s) = c t se(c) = s,.

For the two extreme values of c, O and 20,~ the resulting estimators, the MLE and

the James-Stein estimator respectively, are minimax. It is still not known whether or not

any other value of c results in minimax estimators. However, the limited translation EB

rules are shown to have a minimax property in a certain averaged sense. To be precise,

let up,r denote the uniform distribution on the p-dimensional sphere of radius r. The

estimator 8 of 8 is said to be spherically minimax if


r(Up,r, 8) = E{L(0, ))} < p (1-20)









for all r > 0, where the expectation is taken with respect to the conditional distribution

of X given 8, Xl |9 ~N,(0, I,), and the uniform distribution for 8, O ~ up,,. If the risk

of an estimator as a function of 8 depends on 8 only through ||9||2, Which is the case for

the M'LE and the Jamets-Stein estimnator, then r.(21,, II ,) = R(0, a) so that the ideas

of spherically minimax and minimax coincide. For other estimators r(up,r, 8) gives the
average risk over ||9|| = r.

The limited translation EB estimator is spherically minimax. Its average risk over a

sphere of radius r = ||9|| is given by


7("p,Ipp LEB)=;1[ __ Sc scS(T2/2)], (1-21)

where


p-2 p-22i) 1)

and K ~ Poisson(A). The spherical risk of the limited translation rule is less than or equal

to p since the bracketed term in Equation 1-21 is less than one.

The risk of OfEB, .S 8funCtiOn Of 8, iS HOW examined. It is shown that the risk of the

limited translation estimator of Os depends on 0 through Of/||9||2 and ||9||2. We denote

this risk by


R(0s, OfEB) =Eg(04 OfBL" 2i p"~s ). (1-23)

For fixed p, O < c(s) < 20- and ||9||2, p,c,(s)(. 2)" is an inCTreSing function of its

first arg~umentI. It is therefore~ maimhllized for Of/|||| =VCc,,,,eT ,c~,C, ,,,,~Ca ~al 1. That is, the most unfavorable

case for the estimation of Os occurs when 04 = 0, j / i. In this case the vector parameter

8, for real valued w, has the form 8 = w(0, 0, .. ,0, 1, 0, .. ., 0)T and | |9| = |W| and thus

the risk function becomes R(8i,0 BEB) = fp~c(s) 1, 02). As a function of |w|, fy,c,(s(, W2>

increases from its minimum of 1 (p 2)se/p at w = 0 to a maximum exceeding 1 and

thereafter decreases .-i-mptotically to 1 as |w| increases. Let 8,(s) be the value of |w|









which maximizes fy,c(s) (1, w2). Then, the supremum over all 8 of the risk of OfEB, Which

we will denote by R(p, se), is


Rip, se) = sup R(0s 6, 0{EB) p ,~c( 1 (s) p 2). (1-24)


Equivalently, R~p, se) is the supremum of the risk of OfEB over all priors ( on 8


Ri(p, se) = sup r((, 0{EB). (1-25)

For all values of p, the N----_~- -1 reductions in R(p, se), as a function of s,, occur near s, = 1.

Thus R(p, se) can be considerably reduced by increasing the Bai-; a risk of the Bayes rule

by very little.
We now drop the assumptions that the variance of the sampling distribution, a2

and the mean of the prior distribution, p, are known. We consider independent normal

measurements Xij| I if N(0sla2), j = 1,2,..., k, while 04 are themselves normally

distributed variables 04 IEt N(p, -2), i = 1, 2,...,p. In this case the Ba-, a rule for

estimating 04 is given by


0," = X, i = Xi B(X, p), (1-26)
1+ (k-r2 l2

where X4 = k-' CE Xsy, A1 = (kr~2 /2 and Bi = (1 + A1)-'
Recall that the limited translation Bai-; a rule follows as closely as possible the

Bai-; a rule without however allowing deviation from the MLE bigger than a fixed value

m, = n-. That is, we impose the restriction that |Xi 0"| < m which is equivalent to

|Xi p| 5 mB-l. This rule can be briefly written as


ef" = xi Bpe {B(X4 p)2)( -~

=- p [1 Bpe {B(X4 -1 )2 _p_27)

The unknown parameters p, a2 and -r2 need to be estimated from the data. First, for

k > 1, the conditional distributions of the Xij given Os, j = 1, 2,. ., k and i = 1, 2, .. ., p,










give us


pk
2 X )2(-) (1-28)
i= 1 j= 1

Thuls, letting S = X,-X)2, it fO 10WS that, (2 =S/[p(k- 1)] is unbiased
estimator of a2

Marginally, Xi ~ NV(p, a2/k + -r2) = V~2/(kB)). Hence


V = (X X2 2(kB/_, (1-29)
i= 1

where X = (pk)-' CEC,= Xe. It follows thait for p > 4


(p 2 (1 30)

We now write the unknown B that appears in the formula of the limited translation Bai-i



a2 B
B =.(1-31)
k a2/k'

An unbiased estimator of the first term is 0-2/k while an unbiased estimator of the second

term is (p 3)/V. Also note that the unknown prior mean p is estimated by X.

Replacing the unknown parameters by their estimators results in the following
estimator


0 E = X +' 1x )~ )2 ( pca (Xi X). (1-32)
kV kV

The properties of this estimator are examined and it is shown that it has slightly 0 r-i

Bai-; a risk that the regular EB estimator but it protects the statistician against large

frequentist risks.









CHAPTER 2
MULTIVARIATE LIMITED TRANSLATION BAYES ESTIMATORS

In this chapter we develop robust estimators of the multivariate normal mean

assuming that all the model parameters are known. The cases of some or all parameters

unknown will be addressed in later chapters. The organization of the sections of this

chapter is as follows. In Section 2.1 we review some facts concerning the multivariate

Bai-;- estimators. In section 2.2 we introduce the limited translation estimators. In

Section 2.3 we evaluate their Bai-; a risk performance under the assumed prior. In Section

2.4 we evaluate the frequentist risk of both the regular Bai-;- and limited translation rules.

In Section 2.5 we compare the Bai-; a risk performance of the two competing estimators

when the assumed prior departs from the true prior. In Sections 2.4 and 2.5 we consider

a specific form of prior, the g-prior originally introduced by Zellner (1986). Some of the

long algebraic derivations are provided in the Appendix A.

2.1 Bayes Estimators

We begin by considering the following scenario. The p-dimensional random vector

X, conditional on 8, has the multivariate normal distribution Xl |9 ~N,(0, E), where

E = E{(X 8)(X 8)T}. The mean vector 8 itself follows a multivariate normal

distribution 8 ~ ( NV,(pw, A). Here we assume that all the model parameters, C, pw and

A, are known and the statistician is interested in estimating the unobservable 0.

The assumptions stated above imply that conditional on X = z, 8 is distributed as


8 | X = z ~ N(II -- E(A + E)- (I1 pU), E E(A + E)- E). (2-1)


Thus, considering the matrix loss function


Ll(0, a) = (8 a)(e a) (2-2)


the Bai-;- estimator of the unobservable 8 is given by the posterior mean


8" = X E(A + E)-1(X pw) = (I, B)X + Byw, (2-3)









where B = E(A + E)-l and II, denotes the identity matrix of order p. The B is-;- risk of
the B is-c 4 estimator is given by the posterior variance-covariance matrix

ri((, 0 ) = ~~~E{(( 0")(8 0 )"}=(I BE 24


which is smaller than the B is-c < risk of the maximum likelihood estimator (j!1.1 ),

ri (l, X )= E .
Much in the spirit of Efron & Morris (1971), we now calculate the B is-c 4 risk of the
B is-c 4 estimator 8B for a normal distribution with mean vector pw, and variance-covariance

matrix A,. Let (4 Nz,(pw,, A,). Then the true posterior mean is given by Of

(II, B,)X + B,p.,, where B, = E(A, + E)-l. The B is-c < risk of OE under (* is

ri(Ec, 0 ) =E{L1(8 0 )} =) E{(( 0"~)(8 0"~) }

=E{(( OfH + O d0 )(8 Of" + O 0~") }

=E{(( Of )(B Of ) } + E{(OfP 0"H)(Of 0)") }, (2 5)

where E{(( Of )(8 Of )T} = (II, B,)E. We also have that

E{(Of 0")(Of 0") }

= E (II, B,) X + Bx p, (II, B) X -B

x {(I, -B,) X +B,p, I-B) X -By }"`

=E [{(B B,)(X p,) + B(p, p)} {(B B,)(X p,) + B(p, pU)} ]

=(B B,)B,1E(B B,)T + B(pU, pU)(pU, pU)TBT. (2-6)

Thus, the B is-c 4 risk of the B is-c 4 estimator under misspecified priors can he expressed as


rl(l, di ) = (17, B,)E + (B B,)B,1E(B B,)T + B(pU, rU)(rU, pU) B (2-7)

It is clear that when the true prior mean, pw*, is far from the assumed prior mean, p., the
B is-c 4 risk of 8B is quite high.









Furthermore, the frequentist risk of 0", denoted by R1(0, 0"), is calculated as


R1(0, 0")j = Eg (0"f 8)(0"i 8j)}

=Ee[{(I B)(X ) +B(ru- 0)}{(I B)(X ) +B( 0)}"]

=(I, B)E(I, B) + B(pw 8)(pw 8)TB. (2-8)

It is now easily seen that if pw is far from 8, the frequentist risk of 0" is quite high, the

reason being that OE shrinks the MLE of 8, 8o = X, towards the prior mean pw.

On the other hand 8o = X has minimax risk equal to R1(8, 8o) = E for all 8 and

thus the Bai-; a risk of 8o is also equal to E which, however, is N r;-i than the Bai-; a risk

of the Bai-;- estimator, rl((, OS) = (Ip B)E, if the assumed prior ( is the 'true' prior.

2.2 Limited Translation Bayes Estimators

The multivariate limited translation rules are akin to the Bai-; a rules but at the same

time they put a limit to the amount of shrinkage of the maximum likelihood estimator

towards the prior mean. The goal of these estimators is to maintain low Bai-; a risk and at

the same time put a bound to the frequentist risk.

Suppose now that X z, i = 1,. ., n, are independently distributed NV,(0s, E). Also,

assume that 8i are iid NV,(pw, A).

Definition For the ith vector 8i we define the limited translation Bai-;- estimator of

maximum translation c as


= i X, E(A + E) hc {(A + E) z (Xi p)}, (2-9)

where


Ac(z) = z min(1, c/|| z||i), zE sJ (2-10)


is the multidimensional Huber function, Huber (1974), and c is a known constant.









Equivalently, the limited translation estimator can be written as


., =X, -B(X, I-ipc(||I(A + E) a (Xi Cl)||2), (2-11)

where


pc(u) = min(1, c/ ~), a E R (2-12)


is termed the relevance function, Efron & Morris (1971, 1972a). The limited translation

rule can also be represented as a weighted average of the maximum likelihood and the

Bai-;- estimators since




+{(I, B)Xi + Byw~pc(||(A + E)-E(Xi pw)|| 2). (2-13)


Marginally the random vectors Xi ~ NV,(pw, A + E). Thus, the argument of the relevance

function is the standardized squared norm of Xi, ||(A + E)-i(Xi pw)||2. The value of

the relevance function decreases with the increase in the value of the standardized squared

norm of Xi, thus reflecting the idea that the relevance of the population parameters, pw

and A, is not the same for all 8i. When the observed Xi has a high standardized squared

norm, the Bai-;- estimator, and implicitly the prior parameters, is not considered to be

very relevant for the corresponding 8i. In such a case the MLE is considered to be more

relevant and thus the shrinkage towards the prior mean is appropriately controlled.

In the subsequent sections of this chapter we will drop the suffix i and work with a

generic Of"B

2.3 Bayes Risk of the Limited Translation Bayes Estimators

First, it is of interest to know how well the estimator OfL performs assuming that

the normal prior NV,(pw, A) is the true one. We thus calculate its Bai-, risk, rl((, Of"B)=

E{(( Of"B)(8 Of"B))}. The calculations for the most part do not depend on the choice

of the relevance function pc(.). The following Theorem shows that the Bai-; a risk of the









limited translation estimator can be written as a weighted average of the B aves~ risks of the

MLE and of the Bayes estimator.

Theorem 2.3.1. For r,:; relevance function pc(.) we have


rl((, 8," ) = r,((, X)(1 se) + rl((, 8 )sc, (2-14)

where 1 s, = E[1 pc(U)]2 8811 U >i+2*
Definition For an estimator 8 of 8 the generalized relative savings loss of 8 with respect
to X is defined as


GRSL(0; X) = [rl((, X) rl((, 0")]- [rl((, 8) rl((, 0")]. (2-15)

The term rl((, X) rl((, 0") is the savings, in B i-;-s risk sense, that occur when using

the B aves~ estimator instead of the MLE, while rl((, 8) rl((, 0") is the loss that occurs

when using 8 instead of the B aves estimator.

The generalized relative savings loss of Of"B is given by

GRSL( OL; X) = (1 se)-fp, (2-16)

and for the special case where pc(u) = min(1, c/ ~), 1 sc is given by

Di,2 ( ) C2
1 s, = ?(@>+2 > C2) C >+1 >2 C2)+ _P >2 C2). (2-17)

Proof. We first write

r1(, f =E{((6 Of ~)(e Of ") })

=E{(0 0"+0 "H- Of)( -0" + 0" -0 dk) }

= E(( 0")(8 0")T} + E{(0" OfL")(6 -B O")"}>, (2-18)

where the cross product terms do not appear since E(0|X) = 0B and thus


E{(( 0")(0" Of"") } = E{E(e 0")(0" Of ) |X} = 0. (2-19)









Also, since 0" is the posterior mean, it follows from 2-1 that


E {(( 0") (8 8 ) } = (Is, B) E. (2-20)

Wenow need,, tocalculate E{(0"- O f")(6B Of"))}. Note that


0" Of"" = B(X w) {1 pc(||(A + E)- (X pw)||2i)}, (2-21)

and we thus have


E{(0" Of")( 0" Of") }

=BE[(X p)(X p) {1 pc(||I(A E )- (X pj1)||2 2 T. (2-22)

Let Z = (A + E)-a(X pw) ~ Ns,(0, Is,), and it follows that

E(0" Of"L)(0"B OfL") = E(A + E)-iE{ZZ [1 pc(||Z||2 12}(A + E)-dE. (2-23)

The following lemma simplifies the calculation of the B n-;-s risk.

Lemma 2.3.2. Consider the random vector Y ~ Ns>(0,,721p). TChe the TradOM SCaltT

||Y||2 Gnd the TradOM matri.:YT IY 2 aTC :1,:./. If :. 1,:.1,'/// diStributed.

Proof. First, ||Y||2 iS COmplete and sufficient for -r2. Noting that -r-]TY ~ N,(0, I,), the
statistic Y Y /| |Y ||2 is aECillary. NOW the independence of | |Y | |2 and Y Y /| |Y ||2 follows
from the well known theorem of Basu. O

We now continue with the calculation of the B n-;-s risk. By Lemma 2.3.2

DIZ''1-,( IZ~)2~ 0(ZZT




Again by Lemma 2.3.2, we have

D(Z' (ZZ I' s ZZT" ( "_ 25
E(Z )=E | |2 |||2 = | |2 E || |2) 2-5









and thus


(Z EZZ p- I (2-26)
||Z ||2 E ( ||Z ||2 9zz

Since ||Z||2 2 i~

E{||Z||2[ c,(||Z||2 2} = E{Y[1 pc(Y)]2)

=~I [1' pc()2 exp(- ) dy pE [1 pcll~(U/)]2} (2-27)

where U ~ X>+2. It follows from Equations 2-24, 2-26 and 2-27 that

E {ZZ [1 pc(||Z|| ")2} = E{[1 pc(Ui)]2}I,. (2-28)

Equations 2-18, 2-20, 2-23 and 2-28 show that

rl(l, iiL) = E-_BE[1-_E{1-_pc(U)}2]

S EE{1 p(U)}2 +( BE)[1- E{1 p(U)}2]

= (1 se) + (Ip )sc

= r ((,X) ( e) +T1(C 0 )Sc,(2-29)

where 1 sc = E{1 pc(U)}2.

NUow, by choosing- pc(U) = mnin(1, c/ ), wh~er~e U' ~ \i>+2, We halve

E{1 pc(U)}2 = E{1 min(1, C 2

=E{(1 -)21 0 > C2)

=E{I(U > c2)} 2cE{U ~I(U > c2) C2E{U- I(U > c2)

=P(\i>+2 > C2j C >+1 > 2 -1C2 x, C2, (30
r(2~

which, for fixed p depends only on c and it is independent of the model parameters.
Further, if we feel that the B we;- rule is irrelevant for observations that have
standardized norm bigger than some value co ;?i, co > c, we can modify the relevance









function in the following manner:


p (U) min(1, c/ )> if U < c (-1
0 if U > c .

The cost of this modification in terms of increased generalized relative savings loss is


E{1 p (U)}2 E{1 pc(U)}2

= E(1 C21(C2 < U < c)}+E{(U > c)}-{(- C 21C
= P(U-c)-E{(1- C >CU)

= c{ (U > c )} c2E{U- I(U > c))}

= cE( P(U > c )~ p C2P-1(U > c ). 2-2

~(2~

We have seen that the Bai-; a risk of the limited translation rule is a weighted average

of the Bai-; a risks of the Ba a~ rule and the MLE, the weights being s, and 1 s,

respectively, which causes a generalized relative savings loss of (1 se)Ip. Also, the weight

of the Bai-; a risk of the MLE, 1 s,, for fixed p, is a decreasing convex function of c. This

allows the statistician to choose c by deciding by what proportion it is worth increasing

the Bai-; a risk of the Bay, a~ rule in order to receive protection against large frequentist

risks.

2.4 Frequentist Risk of the Limited Translation Bayes Estimators
We~ nowtur n ou~ r attention to the freque~nti st riskr of Of", a function of 8 denoted by

R1(8, Of"B), to show that the limited translation rule, in return for the increased Bayes

risk, does not allow the frequentist risk to be very large, in contrast with the Ba-, a rule.

The calculation of the frequentist risk of the limited translation rule was feasible only

under the simplifying assumption that the prior variance-covariance matrix is a multiple of

the sampling variance-covariance matrix, that is A = gE where g > 0 is a known positive

scalar. That is, we consider the case where Xl |9 ~N,(0, E) while the prior distribution is









taken to be 0 ~ ( NV,(pw, gE). Such priors, originally introduced by Zellner (1986), are
called g-priors.

Under the assumed model, B reduces to B = (1 + g)-1I, and the Bayes estimator is

given by OE = X (1 + g)-1(X pw). The frequentist risk associated with it is obtained
from Equation 2-8,


R1(8, O ) = g2( g-2C + 1+ -2(~ _)r 8T. (2-33)

Also, the limited translation estimator is is given by

Of"" = X (1 + g)- (X pw)pc(||(1 + g)- E ~(X pw)||2). (2-34)

We would like to compare the frequentist risk of the Bwes;- estimator to the frequentist

risk of the limited translation estimator. An expression of the latter is provided by the

following Theorem.

Theorem 2.4.1. Under the assumption that A = gE where g > 0 is a known scalar and

for the relevance function pc(u) = min(1, c/ ~), the frequentist risk of the multivariate
limited translation rule is given by


Ri(0, Of"") R1(0, 0")

+(e )(e6'( )(1+2)(1 g )-2 ~y +4()r( C2

-2(1 + gc)- P[ ~+2 ) > C2( ~


+2c(1 + g) 1Ex(?~,X 1 { ~ y, [X 2 C2(1+~jl

-2c(1 + g)-t E { [?+aX1 I 4X X +4 C2(1$~1
+ 1+g(1g- 2 C2

+C2( -1E { [ 2 -1 2 C2,( 1 g

-2c(1 + g)- Ex(?~,X 1 { ~ y, [X 2 C2(1 g C)]}, 2-5









where A = (8 pW)TE- (8 p)/2 and "2 (A) ""~" deoe a" non-cntra chi-qur variab.,,,le with

non-c. nhodH ol~ parameter A and degrees of freedom k.

The proof of the theorem is given in section A.2 of Appendix A.

Since the risks in Equations 2-33 and 2-35 involve matrices, in order to graphically

compare them, we consider scalar versions of them. Specifically, we consider the quadratic
loss function


L2(8 _)=( aT -1(8 a,). (2-36)

It is easy to show using Equation 2-33 that the risk of the Bai-; a rule under the loss

function L2 is equal to


R2(e a" ) = Eg ((6 )ej'- (e a )} = tr[E-' R1(0, 0")]l = 22.(2-37)

The following Corollary provides an expression for the risk of the limited translation Bai-i

estimator under the loss function L2-

Corollary 2.4.2. Under the loss function L2 the TiSk of the limited translation B.r;,e

estimator is given by


R2( 8, LB 2 K~81 B" +2,- z(), >' C2 2pq 4A
(1 g)2 1 +
+2A(1 + 2g)(1 + g)-2P[, 4X > C2 1 + )l C2( g-17 ,,() > C2( )

+2c(1 + g)- 2A {[ 22 >C




The proof is given in section A.3 of Appendix A.

The bracketed term in the last two lines of Equation 2-38 can be calculated as

e- Ak Lp +1+2k 2aX
k=0k a( ) p +2

The risk of 8 ,LB for fixed p and g, quite conveniently, is a function only of the non-centrality

parameter A = (8 pW)TE- (8 p)/2 and so is the risk in Equation 2-37.










Let us now consider the hypothetical scenario where the statistician is given n

observations of dimension p = 3. Also suppose that g = 2 and that the statistician is

willing to have a generalized relative savings loss of 1 s, = 101' in order to receive

protection against large frequentist risks.

In Figure 2.5 we see how 1 s, decreases as c increases for three different values of

p = 3, 5 and 10 and for fixed g = 2. For p = 3 and 1 s, = 10I' the corresponding value of

c is 1.52.

In Figure 2-2 we see how the risks in Equations 2-37 and 2-38 behave as the

non-centrality parameter X increases. For small values of A, i.e. when 8 is close to the

population mean pw, the Bai-; a rule has slightly smaller frequentist risk than the limited

translation Bai-; a rule. However, the frequentist risk of the Bai-; a rule increases linearly

with the non-centrality parameter which clearly means that the Bai-; a rule has high risk

when the 8 is far from pw. On the contrary, the frequentist risk of the limited translation

Bai-; a rule becomes flat after A exceeds a certain value. That is, the limited translation

rule does not allow large frequentist risks even if the unobservable 8 is far from the prior

mean pw.

Returning to Equation 2-38, we write R2 8, 8LB) 2 R(, 8B) + p,c,g (A). The

proposed estimator 8,LB does better than the Bai-;- estimator when the function ee,~,,(X)

takes on negative values. This, in general, happens when attempting to estimate a

random effect 8 which departs widely from the assumed prior mean pw, that is, when the

non-centrality parameter A takes on large values. The questions of interest are what values

must A take, for fixed values of p, c and g, in order for the function ee,,(A) to become

negative, and how likely those values are.

We attempt to partly answer this question by providing in Tables 2-1, 2-2 and 2-3

the minimum values, k of A needed in order for ep,c,,(A) to take negative values, for fixed

p, c and g. We also provide the probabilities that A takes a value as big or bigger than k.










These probabilities are calculated assuming the the prior I is the true one, i.e.


P(A > k) = P[(B pU)TE- (8 p)/2 > k] = P(X2 > 2kg-1). (2-40)


Table 2-1 shows the values k and the corresponding probabilities P(A > k) for the

case where the dimension is p = 3, for five different values of the prior parameter g and for

three values of c. We may recall that c and 1 s, are one to one functions and thus the

Table provides the generalized relative savings loss, 1 s,, along with the corresponding c.

Observing the first row of Table 2-1, it is clear that for all values of g, P(A > k)

is bigger that 1 the generalized relative savings loss. That is, by sacrificing 1 of the

Bai-; a risk, we have fairly big returns in terms of the frequentist risk. Similar are the

results di11l-plw I on the second row of Table 2-1. The generalized relative savings loss is

5' while the returns in frequentist risk are bigger than 5' for all values of g. For the case

where 1 s, = 101' the returns in frequentist risk are '? r-i than 101' for g = 2, 5 and

10 and smaller than 10I' for g = 0.5 and 1. This, however, is not discouraging because

the reported percentages, P(A > k), are calculated assuming that the prior ( is the true

one. We can expect the probabilities P(A > k) to increase with the increasing distance of

I from the true prior.

The results of Tables 2-2 and 2-3, where we have chosen p = 5 and p = 10

respectively, are similar. We have fairly big returns in frequentist risk when sacrificing

1 s, = 1 and 5' of the Bai-; a risk. The returns in frequentist risk when sacrificing for

1 s, = 101' of the Bai-; a risk are bigger that 101' for g = 2, 5 and 10 but they are smaller

than 10I' for g = 0.5 and 1.

2.5 Robustness of Limited Translation Bayes Estimators

In this section we investigate how well the limited translation Ba-, a rule performs

when the assumed prior deviates from the true prior. We consider the same model as in

Section 2.4. That is, X|9 ~ Nv,(0, E) and 8 ~ ( -- N,(pw, gE).









Under the assumed model, the Bai-;- estimator, namely the posterior mean of 8 given

X, is 8" = X (1 + g)-1(X pw), and under the matrix loss function L1, given in 2-2,
its Bai-; a risk is


rl((, O ) = g(1 + g)- E. (2-41)


Under L2, the quadratic loss function of 2-36, the Bai-; a risk of the Bail rule is


r2(I B" = rgl+ -1 -1C]=p~ g-1. (2-42)


Now suppose that the true prior is 8 ~ ~* NV(p*, g*E). Then the BEi;- risk of OE under

the L1 and L2 l0SSeS is


ri((* O )i( = (g2-2 ~ f 1+ -2 a* s)C1 TU2, (2- 43)

and


2 BI* __) = 2 g) )-2 + 1+g-2(w )T -1(W a

= p(g2 g*) 2A(1 g*)}(1 + g)-2, (2-44)


respectively, where A = 2-1(1 g*)- (pw pW) E 1(p* pU). When A = 0 and g* / g,

that is when the prior mean has been correctly specified but g* has been over or under

estimated, the Bayes risk r2~ 8B), OVeT Or under estimates the true Ba-, a risk, r2(* 8B)

Also, T2~* 8B) inCTreSeS linearly With A and thuS T2~ 8B) underestimates the true Bai-i

risk when the prior mean is misspecified.

We now consider the limited translation Bai-; a rule. Under the assumed model the

limited translation estimator is given by


=," X (1 + g)-1(X pw)pc(||(1 + g) z Ed(X pw)||2), (2-45)









and using the result of Theorem 2.3.1 it is easy to see that


rl((, 8(") = E(1 se) + E(1 -1)Se (1 sc)E, (2-46)
1+g 1+g

which implies that

r72 ~ 8LB) p -1( Sc) = 9 Sc/(1 + g)}. (2-47)

The following theorem provides an expression for the Bai-; a risk under prior I* of the

limited translation Bai-; a rule obtained under the assumed prior (.

Theorem 2.5.1. For the relevance function pc(u) = min(1, c/ ~), the B.r;, i. risk under (*

of the multivariate limited translation rule obtained under the assumed prior ( is given by




+(1 + g)-2(~ _)C1 T u (g* 2g 1)(1 g*) 1P[y2(X 2 'd

2dP[ i+2(X > C2cl C'dE{ [ g+4 ()-1 ?i+4( > C23



+(1 +g)- -(g* -2g )(1 g ) P[ +2 C2,"
+cE~"" c[p- 2 d [2 2 Cd] (248


where d = f (g, g*) = (1 + g) (1 + g*)-

The proof is given in section A.4 of Appendix A, while in section A.5 we prove the
following result.

Corollary 2.5.2. For the loss function L2,th B 8.11/. TiSk under (* of the multivariate

limited translation rule obtained under the assumed prior ( is given by


r.2 LB*,~~) 7.2(C BR (q ~ -1 2)}(1 + g) IP[X +2(X > C2d

-2A(g* 2g 1)(1 + g)-2 41 ,X > C2d] C2( g- 12 )` > C2d

-2c(1 +g) 2(1 + g*)- E ([ I(A)] I[X (A) > c2d]

+4cA(1 + g)- (1 + g*)-~ E { [ g+2~~- +2~, ( C'd] (2-493)










The last two lines of the above expression are calculated as

3 1 1 -A k +12)2A
25 c(1 + g)- (+g)-22 +12 k >[ C~%d] { 1. (2-50)
k=0k. F( ) p + 2k

We now revisit the example of the previous section where we supposed that we are given

n observations of dimension p = 3 and based on our prior beliefs we set g = 2. The choice

of c = 1.52 corresponds to generalized relative savings loss of 1 s, = 101' When the

prior parameters are correctly specified, the Bai-; a risk of the Bai-; a rule and of the limited

translation rule are T2~ 8B) = 2 and T2(~ 8LB) = 2.1 respectively.

In Figure 2-3 (a) we plot the risk functionS T2~* 8B) and T2~* 8LB) foT ValueS of the

non-centrality parameter A ranging from 0 to 15 and assuming that g* = g = 2. In the

same graph we plot r2~ 8B) and T2( ~ 8LB) Which, however, do not depend on A. We see

that for very small values of A, the Bai-; a rule has smaller risk than the limited translation

rule. However, the Bai-; a risk of the Bai-; a rule increases linearly with A while the Bai-i

risk of the limited translation rule increases in a much smaller rate.

In Figure 2-3 (b) we plot the same four risk functions for values of g ranging from

0.2 to 10 and for A = 0, that is assuming that the prior mean is correctly specified. We

see that for values of g close to the true value, g* = 2, r2(* 8B) is leSS than T2~* 8LB)

However, when g* is underestimated the limited translation rule does better that the

regular Bai-;- estimator. As the assumed value g, of g* becomes bigger than the true value

of g*, the Bai-; a risk performance of the two estimators becomes similar. As g increases,

the two estimators become closer to the MLE and their Bai-; a risk tends to the Bai-; a risk

of the MLE, r2((*, Xi) = p, and here we have taken p = 3.





































0 1 2 3 4




Figure 2-1. Plot of 1 -s, as a function of c.






R2(l LB








o




0-


p= 3
p= 5
p= 10


0 110 210 310


40 50 60


Figure 2-2. Plot of risk as function of the non-centrality parameter, A.


















c= 2.466 1.79 3.81 8.27 22.14 44.94
1 sc=1 .1I 5.45' .7 31: 2.95' .
c=1.840 1.72 3.32 6.72 16.96 33.60
1 -sc= 7/* 8. :'. 8 1 1 7.91 8 1
c=1.521 1.76 3.21 63.23 15.21 29.84
1 sc 10' 7.01 .' 9.2' 1' 00 0.1'. 1 .


c=2.806 2.49 5.12 10.80 28.31 57.11
1 sc=1 7.6 !' .7 5.55' 4.5;:' .1
c= 2.144 2.51 4.69 9.24 22.88 45.19
1 sc= 5' 7.41 9. Is' 9.91' I'. 1 .2 1 .1
c=1.797 2.633 4.638 8.88 21.38 41.98
1 s = 0'. 6.1'7' 9.55' 13 '. 28: 13.5'7'


c=3.490 4. 17 8.24 163.82 42.98 86.05
1 sc=1 .5 .1'. 7>! .0 6 '**'
c= 2.757 4.41 8.03 15.42 37.57 74.21
1 sc= 5' .;'. 9 7 '. 1 3 1 '. 1 .'I
c=2.352 4.77 8.33 15.53 37. 10 73.01
1 sc=10' 39 :' 8.2_1' 13 '. 1 .5 '. 1 2


Table 2-1. Minimum values, k,


of A and P(A > k) for p


Table 2-2. Minimum values, k,


of A and P(A > k) for p
R


of A and P(A > k) for p
R


Table 2-3. Minimum values, k,















-- LB True Prlor
S- Bayes True Prlor
LB Assumed Prlor
Bayes AssumedPro



















0 5 10 15








-- LB True Prlor
S- Bayes True Prlor
LB Assumed Prlor
Bayes AssumedPro


0 10 20 30 40 50


9b



Figure 2-3. B is-c < risk under misspecified priors. 'LB True Prior' refers to T2~* 8, LB)
(B8i1l True Prior' refers to T2 La pB), (LB Assumed Prior' refers to
T2(~ Be LB), 'Bayes Assumed Prior' refers to T2 Le pB). (a) B~is-c risk as
function of the non-centrality parameter X and (b) B is-c < risk as function of g*.









CHAPTER 3
MULTIVARIATE LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS: THE CASE OF UNKNOWN PRIOR MEAN

In this chapter we develop limited translation estimators assuming that the prior

mean is unknown but both the sampling and the prior variance-covariance matrices

are known. The organization of the sections of this chapter is as follows. In Section

3.1 we briefly review the Bai-;- and empirical BEi-;- estimators as well as the notion of

influence functions. In section 3.2 we introduce the limited translation estimators and in

Section 3.3 we evaluate their Bai-; a risk performance under the assumed prior. Section

3.4 evaluates their frequentist risk performance while in Section 3.5 we compare the Bai-i

risk performance of the two competing estimators assuming misspecification of the prior

distribution. Some of the long algebraic derivations are provided in the Appendix B.

3.1 Bayes, Empirical Bayes Estimators and Influence Functions

We consider the B li-, Io example of estimation where the a random vectors Xi,

i = 1,. ., n, are independently distributed according to the p-dimensional normal

distribution Xle|9 imi,(0s, E), where E is known. Also, the Os, i = 1,..., n, are iid

according to Osi __ N ,(pw, A), where A is known. The interest is in estimating the Os

under the matrix loss function L1(0s, ai) = (8i ai)(8i ai) .

First assume that pw is known. The Bai-;- estimator of 8 that is the estimator that

minimizes the posterior risk, is given by the posterior mean of Os given Xi = zi. In order

to find this estimator we calculate the posterior distribution of Os given Xi = zi


f (04|Xi) oc f (Xie|9) f(e )

oc exp{- 2- (Xi B ) E 1(Xi 0 ) 2- (Os pU) A- (Os pU)}

oc exp{-2-] [ (E-] + A- )8i 20 (E- Xi + A- pw)]}. (3-1)


It follows that


Os| =zeimiN/(- A- )- (E- xi + A- pw), (E-l + A- )- ]. (3-2)










Noting that (E-l + A- )-l = (E- (A + E)A- )-1 = (A- (A + E)E- )-1, the posterior

mean, that is the Bai-;- estimator of 8i, can be written as weighted average of Xi and yw

smnce


E(04|Xi) = (E-l + A- )-1(E- Xi + A- pU)

=A(A + E)- Xi + E(A + E)- pw = (I, B)X, + Byw, (3-3)


where B = E(A + E)-l. Further, the posterior variance, also the Bai-; a risk of the Bai-i

estimator, can be written as var(0s|Xi) = E E(A + E)-1E = (I, B)E.

Since yw is unknown the Bai-;- estimator, Of = (I, B)Xi + Blw, cannot be used

as such. However, marginally the random vectors Xi have mean pw, and thus the unknown

prior mean yw can be replaced by an unbiased estimator, X, = n-l Cl Xi, thus resulting

in an empirical Bayes (EB) estimator


Of"B= (I,-_ B)X + BXn, (3-4)


of Bi. The same estimator can be obtained as a hierarchical Bai-;- estimator when one

assigns a uniform prior distribution on pw.

The EB estimator shrinks every maximum likelihood estimator (jll.11;), Xi, towards

the grand mean, X,, the MLE of the unknown prior mean, pw. In doing so, it attains

a lower Bayes risk than the MLE, under the assumed prior. However, it results in high

Bai-; a risk when the prior distribution is misspecified. It also results in high frequentist

risk when attempting to estimate parameters, Os, that are far from the grand mean.

On the other hand, the MLE has minimax risk equal to E for all Os. In order to avoid

these two problems, we develop robust EB estimators, namely the limited translation EB

estimators.

We start by assigning the noninformative prior on yw ~ Uniform(sW). We then find

the influence of observations Xi, i = 1,..., n, on the posterior distribution of pw. The

influence is measured using the general divergence formula introduced by Cressie & Read









(1984). Let fl and f2 denote two density functions. Then the general divergence measure

is given by


D (f f2) X-1 I-1 Ef, (fl/f2) -1}. (3-5)

Here, fl and f2 denote the posterior densities of pw given X = (XT,... ,XT)" and
X (-i) = (XT X_,X XT y) respectively. In order to find fl and f2, IlOte that

Xl|pt NV,(pw, E + A), i = 1,..., n, and yw ~ Uniform(W~). It follows that

f(p|lX1,..., X,) oc exp[-2 {p wn(E + A)- pw 2p ~n(E + A)- X,}]

oc exp{-2- n(pw X,) (E + A)- (pw X,)}. (3-6)

We thus have


p|lX1,...,X, ~ NV,(X,, n- (E + A)),

p|IX1,... ,Xi_1,Xi~: ,..X?, ~ N,(X,_zi (n-1)- I(E + A)), (3-7)

where X,_,-L is the average calculated using all the random vectors except the ith one.

The following Theorem provides a result concerning the divergence between two
multivariate normal densities.

Theorem 3.1.1. Let fl denote the NV,(p Ex) 1~:. :7 / and f2 d68016 the 1V(P, 2 2)
1. .1:;l Then


D (ft, f2) __-1 XI-I[1 2i 1
z (C1 z +( 1))~ C1
x exp {(z-p 2 1-
X(2 l i Ii 2) (8

If the variance-covariance matrices C1 and E2 are known, which is the case for our

estimation problem, the divergence measure is a one to one function with


(rw, rW)[2 +XC 2 X1 -1 ~1 2 U). (3-9)









For the special case where the densities fl and f2 are the ones given in 3-7 we
have pi 2 ,_ (n1 1)-1(Xi. X,,) and (1 + A)E2 1 C

(E + A)(n + A)n-l(n 1)-1. It is now easy to see that the divergence measure is a one to
one function with

n(E A)-]
(Xi ,)T (Xi ,)T, (3-10)
(n 1) (n + A

which is a quadratic form in (Xi X,). Based on this result we will obtain some robust

B li-o Io estimators in the following section.

3.2 Limited Translation Empirical Bayes Estimators

A modification of the EB estimator will give us the limited translation EB estimator.

Since the influence of the random vectors Xi, i = 1,..., n, depends on their distance

from X,, in the EB estimator we want to control the standardized distance of Xi

to X,. We thus define D -- var(Xi X,) = (1 1/n)(E + A), and we write

0,9" = Xi BDED-T (Xi Xn).

Definition For the ith vector Os, we define the limited translation EB estimator of

maximum translation c as


0 =e~E Xa BD bc{D- (Xi ,} (3-11)

where


Ac (z) =z min(1, c/|| z||i), zE sJ 7 (3-12)


is the multidimensional Huber function, Hampel (1986), and c is a known constant.

The proposed estimator can equivalently be written as a weighted average of the MLE and
EB estimator since


8k" = Xi B(Xi X,)pc(||D- (X, X,)||2)

= Xe{1- pc(||D- (X i X,)||2)) sl0B c(||D- (X, Xn)||2) 3 )









where pc(u) = min(1, c/ ~) is termed the relevance function.

The limited translation EB estimator follows the EB estimator as closely as possible

subject to the constraint that the distance of the observed Xi to the observed mean X,,

as measured by the standardized norm ||D-T(X i X,)||, does not exceed a certain

value, c ;?i. When this distance takes on a value '?i-.1 r than c, the relevance function

takes on a value smaller than one, and by the second line of 3-13 we see that the limited

translation EB estimator gives the MLE positive weight at the expense of the weight of

the EB estimator and as the distance of Xi to X, increases the less relevant is considered

to be the EB rule for the estimation of the corresponding 8i. In the next sections we show

that this provides the statistician with protection against large values of the frequentist

risk, while slightly increasing the Bay, a~ risk.

3.3 Bayes Risk of the Limited Translation EB Estimators

We now calculate the Bay, a~ risks of the EB estimator, Ty((, 8 ), and the limited

translation? EB estimantor, Ty((, 8 ~f) assum~ing thait the prior ( N~,(pu, A) is th~e true

one. First we calculate


r1(, f" =E{((0, Of")(0, O f") }>
=~ ~ o E{0 Of+OB OEB) (0- oB + OB OEB) }
=~V E{0 Of(0 Of"}+E{O Of ")(Of Of)"


=(I, B)E + BE {(X, pw)(X, pW))}BT

= {I,- (1- n-1B}E.(3-14)

Comparingf riEOf) to rl((, Of), given- in Equatio 2-4, we see that the price for

having to estimate yw from the data is n-1BE which converges to a matrix of zeros as n

increases, the rate of convergence being n-l. This is intuitively clear since as the sample

size increases, the sample mean X, converges to the population marginal mean pw.
The followingr theorem givesp an epvression for r1(( / ( ).B The calcu~latinonsdo not

depend on the special nature of the relevance function pc(.).








Theorem 3.3.1. For r,:; relevance function pc(.) we have


T1(0, ijc )


TI((, X,) (1


se) + T1((, ij )Sc,


(3-15)


where 1 sc = E{1 pc(U)}2 >ihU ~+2*
Hence, the generalized relative savings loss of OfIf" w-ith respect to X,, defined by


GRSL(0 8(f"; X,) =[rl(E,

is c~alc~ulated als GRSL( 0,fif; X;)


X;) r (E, Of H)]- [ry(E, BL J")

=(1 se)lp. If we choose pc(u)


r ((, Of )]


(3-16)


min(1, c/ ~), then


1 s, is given by


C~l >1 C2 -1C2F' > C2)
2 (2 i~ 2


sc P(X >+2 > C2)


(317)


where 0(.) denotes the gamma flinction.

Proof. We write


r1(E, d0, J")


E {(0, O ) L~(Bi 0 ) }Y


=E{((, O f"H+ Of" ~

=rl(E( Of) + E{(Of"


ek, )(e, Of' 8L


d,(,{ ) }>


0 )(Of c'


0 ) }~,'


0,(,{ ) } + E {(O~f '


ijL e(")(e Of"j)}.


(3-18)


Noting that


et7 8,L~ = B(Xi ,) {pc(||D- (X;


Xn>)||2>- _


(319)


it follows, from the independence of X;


X, and X,, and the fact that E( X,)


that


E( {(O 6" -L~ 0 (")(, Of ") }=E EO e" -0"")(0, Of ') |lX;}


BE[{pc(||ID- (X;


Xn)||2) 1}(Xi X)(rW Xn)"]B


0. (3-20)









Next ,


=BE{(Xi X,)(Xi X,)T [pc(||ID- (Xi ,|2_2p

=(1 1/n)E(A + E)-zE {ZZ [pc(||Z||2) 12}(A + E)- E, (3-21)

where Z ~ NV,(0, I,). It was shown in Equation 2-28 that


E{ZZ [pc(||Z||2> 2}" = E[1 pc(U)]21, (322)

where U ~ X@+2. Hence, from Equations 3-18, 3-20, 3-21 and 3-22 follows that


rll, iiE) =T1((, CjE) +E{[1-_pc(U)]2) lnB

=rl(l, e")>+ (1- se)(1-1/~n)BE, (3-23)

where 1 s, = E {[1 pc(U)]2}. The second of the two terms can be thought of as the

price in terms of increased Bayes risk for limiting the frequentist risk of the EB estimator.

Alternatively we can write


rl(c(, efi) = B~s,(1 -1/n) = E(1- se)+ {E (1- 1/n)BE)Se

= 1((, Xi)(1 se) + r1((, CjE)so, (3-24)

thus completing the proof of the Theorem. O

The Bai-; a risk of the limited translation EB estimator is a weighted average of the

Bai-; a risk of the EB rule and the B~i-;-- risk of the MLE, the weights being s, and 1 s,

respectively. This causes a loss in the generalized savings of (1 se)lp. However, the

weight of the Bayes risk of the MLE, 1 s,, for fixed p, is a decreasing convex function of

c. Thus, the choice of c is equivalent to deciding by what proportion it is worth increasing

the Bai-; a risk of the EB estimator in order to receive protection against large frequentist

risks. This protection does not require increasing the Bai-; a risk by more than 10I' .









3.4 Frequentist Risk of the Limited Translation EB Estimators

In this section we focus our attention to the risk of (" as a function of 8 =

/T(B ,0 e)T, which we dlenolte byv R1(0s, 8, "). The purposes is to showz that the limited
translation EB estimator does not allow to the frequentist risk to take large values. The

cnalclation of the riski of 8~~ wa~s possible only under the simplifying assumption
that the population variance-covariance matrix is a scalar multiple of the sampling

variance-covariance matrix, that is A = gE where g > 0 is a known scalar.

It is of interest to compare the frequentist risk of the limited translation EB estimator

to the frequentist risk of the EB estimator. To this end, before providing a result

concerning the risk of the limited translation estimator, we find an expression for the

risk of the regular EB rule. We now write


Rl(Os,l O ") = Eg (0, Of e")(0, Of e") }

= Eg ((O Xi)(04 X,) } + B~: {(Xi X,)(X, ,) }BT

+ Eg((Os Xi)(X, ,) }BT + B~E( (Xi X,)(04 X ) }. (3-25)

Now, Xi x,|9 if (ei e,, (1 1/n)E), where 8, =r n-1 CE Os. It follows that


Ee {(Xi X) (Xi ,) } = E (1 1/11) + (0, Os en(Bi -e) (3-26)

and that the expectation E8( {(0, X,)(X, X,)T} is equal to

040{C ?00 00 () OsO' + E) = -(1 1/12)E. (3-27)
k=1 k=1

Thus, combining Equations 3-25-3-27, we obtain


Rl(s,l Of") = E + 2(1/'1 1)BE + B {E(1 1/nr) + (Bi ,)(0, ,) }B (3-28)

which under the assumption that A = gE reduces to


R1(8,, Of") = (1 + g)-2(8 s- n,(8 s- n 1 (1 + 2g)C ]. (3-29)
1 1/n(1 + g)2









The following Theorem provides an expression for the frequentist risk of 8, ("
Theorem 3.4.1. Under the assumption that A = gE where g > 0 is a known scalar,

and for the relevance function given by pc(u) = min(1, c/ ~), the frequentist risk of the
multivariate limited translation EB rule is given by


R( ( )= g (, O~~ )(0,e O E) } R,(B Of )

+(e, e,) (e, %j) (1 + 2g) (1 )- +42( i > C2

+2(1 + g)- P[y +2 i) > C2( )

~"(+ C2 -1E {( [?i 14 X~-1 i 14 i) > C2(l~/ 1

+2c(1 + g)-~ Ex { [7 +2 +2i~- i[,, ) > C2( gl

-2c(1 + g)- E { [7 +4 f: +4 x i) > C2( ))

+ (1 /n 1 g(1 g)-2lt,(X) +22( ) > C2
-t"l+C g-1E { [X 2 i~-1lx 2 aX) > C2( gl

-2c(1 + g)-f E {[ it2()l i (1+2 4) > C2( i)]


(330)


where As = 2-1(1 1/n)- (0, 8,) E (0, 8 ,) and X (A) denotes the non-central
chi-square distribution with non-c. oh WH 0l~ parameter equal to A and k degrees of freedom.

The proof is given in section B.1 of Appendix B.
For easier comparison of the risks of 3-29 and 3-30, we calculate their scalar versions

by considering the L2 lOSS funCtiOn


L2 8, T)=( a'-( -1 a


(331)


First, it is easy to show that the risk of the EB, rule under the loss function L2, iS equal to


R2 8 sl0B) = p + 2(1 -1 )(+g-2s (1 -1n( g( )2,


(3-32)


which, for fixed p and g, is a function only of the non-centrality parameter Xi, and so is
the risk of the limited translation estimator, as becomes evident in the following Corollary.









Corollary 3.4.2. Under the loss function L2 the frequentist risk of the limited translation

EB rule is given by


R2( 8 L10B) = Eg {(Bi ) E-. (6 )} t[E- RI(i(0, jl)

=R2( 8, s-l-lB i~2 (x, > C2( i p(1 2g
(1 + g) (1 + g)2
+t 2a (1 1/11)(1 +t 2g)(1 + g)-2P[ +4(i C2( j

+ C2( _/2( g-1P 1(L > C2l ~

+ 2c(1-1/ni)(1+g)-a 2AEl~i {[' 2 l, 2x) i+)>C2

-E,{[x (A,)] I[g (A) >, c2~~l]

The proof is very similar to that of Corollary 2.4.2, given in section A.3 of Appendix

A, and thus omitted.
The bracketed term in the last two lines of Equation 3-33 can be calculated as


rd 22)Ek+ +12k >~( C2 X )
k=0k F( )k P + 2k
We now consider the hypothetical scenario where the statistician is given n

observations of dimension p = 3. We also suppose that n is large enough to ignore

the 1/n terms in the risk functions in 3-32 and 3-33. Also suppose that g = 2 and that

the statistician is willing to have a generalized relative savings loss of 1 s, = 101' in

order to receive protection against large frequentist risks. For p = 3 and 1 s, = 1CI' the

corresponding value of c is 1.52.

In Figure 3-1 we see how the risk functions of 3-33 and 3-32 behave as the non-centrality

parameter, Xi increases. For small values of Xi, i.e. when Os is close to Os, the EB
estimator has slightly smaller frequentist risk than the limited translation EB estimator.

However, the frequentist risk of the EB estimator increases linearly with the non-centrality

parameter which clearly means that the EB estimator has high risk when the Os is far
from 0. On the contrary, the frequentist risk of the limited translation EB estimator









becomes flat after As exceeds a certain value. That is, the limited translation rule does not

allow large frequentist risks even if the 8i is far from 8.

3.5 Robustness of Limited Translation EB Estimators

The purpose of this section is to examine the Bai-; a risk performance of the

proposed estimator and compare it to the performance of the regular EB estimator

under misspecified models. This examination proceeds as follows. We first derive EB and

limited translation EB estimators assuming that the sampling distributions of the Xi are

X,|9, iv ,(0s, E), while the 8i themselves are normally distributed, 8i d 1V(wg)

i = 1,..., n. We then calculate the Bai-; a risk of the two estimators assuming that the

true prior is (* NV,(pw*, g*E). It should be noted here that the (mis)specification of the

prior mean does not really matter because we are assuming it to be unknown.

Assuming that the true prior is I, the EB estimator of the ith vector Or is given as

Of" = Xi B(Xi Xn), where B = (1 + g)-l, while its Bai-; a risk under the assumed

prior is given by rl((, Of") = {1 (1 n-1)B}E. If, however, the true prior is (*, the

estimator that we should be using is Of"* = Xi B*(Xi Xn), where B* = (1 + g*)-l

We now calculate the Bai-; a risk associated with the EB estimator derived under the

misspecified model. We have that


rl(l*, Of ) =E{(0, Of") (0, Of" )}

=r z((*, Of "*) + E {( Of"* Of ) ( Of "* Of ) }, (3-35)

where

E{(Of* O ")("* O ) = (B B*)2E{(Xi X )(X, X,) }


= (1 n-1)(B B*)2(B 1 pa -1 n-1 pap2(g 2C (336)

and thus


rl (I*, 00") = r ((*, eEB *) + ( 1 n- 1)B*B2 (g g2C (3









The Bai-; a risk of the EB estimator under misspecified priors has two components. The

first one, rl((*, Of"*), can be thought of as the inevitable risk, the risk due to nature,

while the second one can be attributed to the misspecification of the prior parameter, g*.

As the distance between the true prior parameter, g*, and the assumed one, g, increases so

does the second component of the Ba-, a risk and, of course, the Bai-; a risk itself.

We now turn our attention to the limited translation estimator. Under prior I this

estimator is given by

8cEB = Xi B (Xi Xn) pc (| |D- (Xi X,)11) | | 2


where D = var(Xi X,) = (1 1/n)B-1E. Its Ba a;- risk under prior (* is given in the

followingf Theorem.

Theorem 3.5.1. The Boo., risk of 0 EBI derived r;l,., l mi at G the ETru priOT iS ~, undeT

prior (* is given by


TI((', eL") = TI((',l d B) +(1 1/12)B*E {:(B/Bl*)pe.(U) -1}2 E, (3-39)

where, with c** = (c*B)/B*,


E {(B/B*)pe.(U) 1}2

=[(B/B*) 1]2 +X2 I C*)2] E{( 1)2 0U > (C*)21

=[(B/B*) 1]2Py, +2 (C')2 f [ 2 > (Cr)2
-c**A ) P > (c*)2] +-1 C**)2P[ ? > (Cr)2] (340)


The proof is given in section B.2 of Appendix B.

We now consider the L2 lOSS function, given in 3-31, and provide expressions for the

Bai-; a risks of the two estimators. First, the Bai-; a risk of the EB estimator is given by


T2L EB~ __~~ 1pa_- p 2









while the Bai-; a risk of the limited translation estimator is given by


T.2 L~, fEB ( (1_ -1 B*pa -1n)B*E {(B/B*)pe. (U) 1}2 p. (3-42)


Figure 3-2 shows how the two functions behave as the assumed prior parameter g, varies

around the true prior parameter g*, which, for the sake of comparison, we take to be

g* = 2. We also take p = 3 and a to be large enough to approximate 1 n-l m 1.

When the true parameter g* is assumed to take any value smaller than 1.34, OL~EB

does much better that 8E. That is, when g* is underestimated, the limited translation

estimator has much smaller Bai-; a risk the the EB estimator. When, however, the assumed

prior parameter is close to the true parameter, the EB estimator, as one should expect,

fares better than the limited translation estimator. As the assumed value g, of g* becomes

bigger than the true value of g*, the Bai-; a risk performance of the two estimators becomes

similar. As g increases, the two estimators become closer to the MLE and their Bai-; a risk

tends to the Bai-; a risk of the MLE, r2((*, Xi) = p, and here we have taken p = 3.



















R2 el, 8E)
LEB
-- R2 el>elc )








O


C-










0 10 20 30 40 50 60





Figure 3-1. Risks plotted against the the non-centrality parameter, X .








to r2EB)

LcEB)










(115' -~


0 0 52

Y9


Figure 3-2. B 0-;- risks plotted against the assumed parameter g when the true parameter

is taken to be g* = 2.









CHAPTER 4
MULTIVARIATE EMPIRICAL BAYES AND LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS

4.1 Introduction

Empirical BEi-;- methods are used quite often in the theory and practice of statistics.

Their increasing popularity is attributed to the more efficient inference that they lead

to compared to the classical frequentist procedures. Modelling the similarity among the

individuals or populations leads to this increased efficiency.

In order to make this point clear, suppose that we observe a random variables, each

from a normal distribution with different means but same variance, that is, Xie|8 ~

NV(0s,a2), i = 1,..., n, and suppose that the goal is to estimate the unobservable Os.

The classical approach to this problem is to estimate each of the unknown means using

its maximum likelihood (jlli) estimator, Os = Xi. The problem here is that each of the

random effects is estimated based on only one observation, and it is clear that estimation

based on such small sample sizes cannot be very reliable.

To tackle this problem we model the similarity among the individuals (populations).

For instance, suppose that we observe scores on IQ tests of a individuals. The similarity

among the individuals, in this case, is that they belong to a population the average IQ

score of which is E(04) = 100.

Modelling of the similarity can be achieved by assigning a distribution to the

unobservable Os which leads to hierarchical B li-, Io models. These models can also

be thought of as models for incorporating prior information in the inferential procedure.

For the exponential family of distributions, the resulting estimators, namely the posterior

means, are weighted averages of the ML estimators and the prior means, the weights being

inversely proportional to the sampling and prior variances.

When some or all of the prior parameters are unknown, the Bai-;- estimators cannot

be used as such. However, the prior parameters can be estimated from the marginal

distribution of the data. Replacing the unknown parameters that appear in the Bai-i










estimators by appropriate estimators, results in the so called empirical B?,-c 4 (EB)

estimators.

A well known characteristic of the EB estimators, for the exponential family of

distributions, is that they shrink the ML estimators towards some synthetic mean. In

doing so, they achieve smaller B is-c < risk than the ML estimators but they can lead to

poor estimation of random effects Os, that have unusually small or large values.

Robust EB procedures have been proposed to guard against problems of this type.

Efron & Morris (1972a) developed some robust estimators which they referred to as

'limited translation rules'. These rules are compromises between the EB and the ML

estimators that slightly increase the B is-c < risk but guard against large frequentist risks.

Efron & Morris (1972a) developed limited translation EB rules for the univariate normal

case. The objective here is to develop limited translation EB rules for multivariate normal

case.

One of the virtues of the limited translation rules is that they do not fare too hadly in

their B is-c < risk performance, compared to the regular EB estimators, even if the assumed

prior is close to the true one. In a frequentist risk sense, the limited translation estimators

do not perform too hadly relative to the regular EB estimators even if the random effect

to be estimated is close to the synthetic mean towards which the ML estimators are

pulled. On the other hand, if the random effect to be estimated is far from this synthetic

mean, then the limited translation estimators do perform much better than the regular EB

estimators.

The organization of the remaining sections is as follows. In Section 4.2 we review

some results concerning the multivariate EB estimators. In Section 4.3 we introduce the

limited translation estimators. Section 4.4 evaluates their B is-c < risk under the assumed

prior while their frequentist risk is evaluated in Section 4.5. In Section 4.6 we undertake a

simulation study to evaluate the effectiveness of the limited translation EB estimators and









compare them with the regular EB and ML estimators. The proofs of the results of this

chapter are given in Appendix C.

4.2 Empirical Bayes Estimators

Consider the case where a random vectors Xi, i = 1,..., n, are independent with

X4|9, if N,(0s, E), where E is known. The Os, i = 1,..., n, are iid according to

Os LE ( -- N,(lU, A). Under. thle matr~ix loss function Li1(Os, as) = (ei ai)(8i asi)7, where

ai is a vector guess for Os, the Bai-;- estimator of 8i is


Of = Xi E (A + E)-1(Xi pw) = Xi B(X, pw) = (I, B)X, + Byw, (4-1)


where B = E(A + E)-l. However, Of cannot be used as such when either or both of the

prior mean and prior variance-covariance matrix are unknown.

An EB estimator is obtained by observing that marginally Xi N,"1,(pw, A + E),

i = 1, 2, .. ., n. Hence, the unknown yw is estimated by X, = n-l CE Xi. Also, letting

S = CE (Xi X,) (Xi X,)T, the inverse of the unknown marginal variance covariance

matrix (A + E)-l, is estimated by aS-l, where a is a known constant which we set equal

to a p 2 if we want an unbiased estimator of (A + E)-l. Thus, the resulting EB

estimator of 8i is


Of"" = X, aES-'(Xi x,)j = Xi B(X, X,,) = (I, B)X, + BX,,1 (4-2)

where B = aES-]

The EB estimator shrinks the ML estimator of 8i, namely Xi, towards the grand

mean, X,, the ML estimator of the unknown prior mean, pw. In doing so, the EB estimator

attains a lower Bai-; a risk than the ML estimator, under the assumed prior. However,

it results in high frequentist risk when the Os is far from the grand mean. In order to

quantify the two preceding statements, we calculate the Bai-;- and frequentist risks, under

the matrix loss function L1, of the EB estimator. The following Theorem provides an

expression for the Bai-; a risk of Of", which is denoted by rl((, Of") and calculated as









E(O, ~ aB\ Of)t Of),wt h xettion being taken over the join distribution of

0" = (6T ,.Tf and X"= (XT ..,X().
Theorem 4.2.1. The Ber, risk of the EB estimator, under the assumed prior, is given

by


rl(l, 0") = an-1{2 -a(n p-2)- }BE. (4-3)


The proof is given in section C.1 of Appendix C.

The above Bai-; a risk is minimal when a = n p 2, and with this choice of a, it

becomes


rl(l, Of ) = E n- (n p 2)BE, (4-4)

which increases with p but decreases with n. This is intuitively obvious since as p increases

so does the number of parameters to be estimated. On the other hand, as n increases so

does the information available for the estimation of the prior parameters. Notice that

BE = E(A + E)- E is positive definite. Hence, it is clear that the above Bai-; a risk is

smaller than the Bai-; a risk of the ML estimator Xi, of Os, rl((, Xi) = E.
We now, turn, ourcc, atenio to~ the frequ;ents risk, of~ the EB ;,c, esi ator of Os. This

risk is denoted by R1(8i, Of") and it is calculated by averaging the matrix loss function

LI over th~e sampling distribution? of X, thait is RI(0s, O~f")=E (,O" 4-Oi}

In order to obtain an unbiased estimator of the risk of Of", we use the multivariate

version of Stein's identity provided in the following Lemma.

Lemma 4.2.2. Let h : sP i s be a vector of differentiable functions. Suppose

Y ~ NV,(pw, ). Let hi be the ith element of h and Yj be the je^ element of Y. Then,

if E {|864 (Y) /84|}i is a matrix: with all elements finite, one has CEE {8(Y) /dY }

E {(Y )h (Y7) }
The proof is very similar to the one provided by Stein (1981) for the univariate case
and thus omitted.









The frequentist risk of the EB estimator is calculated as


le~ iEB)j = E,{(0, O f") (0, Of")" }

=Eg [{0 Xi +t B(Xi X,j)} {O Xi + B(Xi ,)}"]. (4-5)

Using the multivariate version of Stein's identity, we obtain another expression for the

frequentist risk of the EB estimator. This expression is given in the following Theorem.
Theorem 4.2.3. The frequentist risk of OfE" can be expressed as


El it, EB) = E + a(aL + 2)CEE( S -(X i X,,)(Xi X,,)7S- }

+t 2aI.E~{Tr{S-1(X i X,)(Xi X,)T}S- }E

2a1 /n)Eg(S-)E.(4-6)

The proof is deferred to section C.2 of Appendix C.

Now, let OEB EB ((~iT, ..:EBT)T anld consider the the loss

L2te 8 EB) = L (6 Of ) = (0, Of") (0, Of") (4-7)
i= 1 i= 1
Under this loss,


R2 8, EBL~) 1i RI(i; EB) = nE {2(nr p) a}E~g(S-1)E (4 8)
i= 1

In particular, for a = n p 2, which minimizes the above, we have that R2 8, 8EB)

nE aCE8(S- )E, and it is clear that R2(8 EBL) 2 C=R(0, X), for all 0. That is,
in the total risk sense of L2, the EB estimator dominates the ML estimator, a result not

surprising in light of the well known univariate analog.
Returning to the result of Theorem 4.2.3, an unbiased estimator Ry (8i, Of ) of

R1(0s, Of") is obtained as follows

1 i, (, EB) = E + a(a + 2)ES- (Xi X,)(Xi X,)TS- E

+t 2a[tr{S-(X, X,)(Xi x,) } (1 1/n)j]ES-'E. (4-9)









This quantity will be large when Xi is far from X,, that is when Xi is an outlier. Among

others, one possible situation when this occurs in a data set is when the true prior is a

mixture of two or more normal distributions, with one of them having high probability.

Under this scenario, it is not very safe, in light of 4-9, to consider the center of the data

X,, to be equally relevant for the estimation of all 8i.

In order to avoid high risks associated with EB estimators corresponding to outlying

observations Xi, and possibly outlying corresponding random effects 04, we develop

limited translation EB estimators that serve as compromise between the EB and the ML

estimators. Since the ML estimators have minimax risk equal to E for all Oi we expect the

limited translation EB estimators also to maintain low frequentist risk and additionally to

maintain low Bwes;- risk.

4.3 Limited Translation Empirical Bayes Estimators

When the prior variance-covariance matrix A is known, the EB estimator is given by


Of" = Xi B(Xi Xn). (4-10)


Based on this estimator, some robust estimators, namely the the limited translation EB

estimators, are obtained by modifying it in a way that controls the amount of shrinkage of

Xi towards X,. This is done by controlling in Of" the standardized version of Xi X,.

Definition The limited translation EB estimator of maximum translation c for the ith

vector Os, is defined as


8" = X B(X,- X,)pc(||Dq~(Xi Xn)||2)


where pc(u) = min(1, c/ ~) is termed relevance function (Efron & Morris 1971, 1972a), c

is a known constant and D var(Xi X,) = (1 1/n)(A + E).

We may note the similarity between the relevance function, pc(u), of Efron & Morris

(1971, 1972a) with the function, Ac(u), of Huber (1974). They are connected through the

equality, Ac(u) = upc(u).









Since the prior variance-covariance matrix A is unknown, we replace (A + E)-l by
aS-l to obtain


0 (E" = Xi B(Xi X,)pc(||k1S- (Xi X,)||2 (11

where kl = (1- 1/n)-2a2.

It can also be written as a weighted average of the ML and EB estimators since


., = X{-p(|Si(Xi X,,I")|| sl0B c(||kS -iX X,||) 12)

The limited translation EB estimator follows the EB estimator as closely as possible

subject to the constraint that the distance of the observed Xi to the observed mean X,,
as measured by ||klS-I(X X,)||i, does not exceed a certain value, c = ar. Wihen this

distance takes on a value '? r-i than c, the relevance function takes on a value smaller

than one, and from 4-12 we see that the limited translation EB estimator gives the MLE

bigger weight at the expense of the weight of the EB rule. As the distance of Xi to X,
increases the less relevant the EB rule is considered as an estimator of Oz.

4.4 Bayes Risk of the Limited Translation EB Estimators

The purpose of this section is to compare the Bay, a~ risk of the regular EB estimator,

rl((, Of"), to the B ii a risk of the limited translation EB estimator, Ty((, BL ),
assuming that the prior ( NV,(p,, A) is the true one.

The following theorem gives an expression for rl(,', 8 (") in terms of rz((,',Xi)

and rl((, Of"n). The calculations do not, depend on the special nature of the relevance
function.

Theorem 4.4.1. For r,:; relevance function pc(.) we have


rl(E, B(")= Ean- LBE[1- E{1- pc(aW)}2] a(n-p- 2)'

-2Elpc(aWr)}{a(n-p-2 )' -1} ( 3

where, W ~ Beta((p + 2)/2, (n p 1)/2).









The proof of the Theorem is provided in section C.3 of Appendix C.

Th~e Bai-, risk of 8c (" is m~linimll for a = n p -2, in? which case

,/ jErl(E, O f~) = E (n-_ ~)2)- B E[1 E{1-- pc(aW)}2] 4)


Letting 1 s, = E1 -, pcm~aW} the expressio for~/ rz((, 0 ") beome


rl(E, e8 ) =T(,X)1-se) + T1((, Of )se. (4-15)

Th~e generanlized relative sain~jlgs loss of e (" With? respecCt to Xi is defineC d as

GR.SL(6 BL; Xl) = [ ((, Xi) r((, Of"B)]- [rl((, ef ") r((, Of )]. (4-16)

The term rl((, Xi) rl((, Of") is the savings, in Bai-; a risk sense, that occur when using
the, EBa estimatorc,, inta fte MLEnrr~, while i(, 0(" -LB r(,f Of) is the loss that

occurs whenl using 80 (" instead of the EBR estimantor.
Hence, for a = n p 2, the ge~neralized1 relatively savi~ngs os of 0 (" With respect to

Xi is is calculated as GRSC L(6 ( Xi) = (1 se)lp*
We now give an expression for 1 s, = E{1 pc(aW)}2 for the choice of relevance

function pc(u) = min(1, c/~). We have that


1 Sc = E{1 pc(aW)}2 = E{1 min(1, cl/a)}2

=E{1 I[cl/aW > 1] (cl/aW)I[cl/a < 1]}2

=E{(1 cl/aW)21[ >-c1 2

=P(Wo > a- c2) C2( l)a)-1 2W > a-1C2)
-2a Fp +1 n +1 _p+2 _n)( >aC_
2 2 2 2

where Wi, i = 0, 1, 2, have the Beta((p + 2 i)/2, (n p 1)/2) distributions respectively.

The Bai-; a risk of the limited translation EB estimator is a weighted average of the

Bai-; a risks of the EB estimator and that of the MLE, the weights being s, and 1 s,

respectively. This causes a loss in the generalized savings of (1 se)Ip. However, the









weight of the Bayes risk of the MLE, 1 s,, for fixed p and n, is a decreasing convex

function of c. Thus, the choice of c is equivalent to deciding by what proportion it is worth

increasing the Bai-; a risk of the EB rule, under the assumed prior, in order to receive

protection against large frequentist risks.

For a = n p-2 and pc(u) = min(1, c/ ~), Figure 4-1 shows how 1- s, decreases as c

increases for two values of p = 2, 5 and two values of a = 10, 30. It is interesting to observe

that for given values of c the smallest values of 1 sc occur for p = 5 and n = 10 while

the N----- -r ones occur for p = 5 and n = 30. An intuitive interpretation of this, assuming

correctly specified priors, would be that when n is small compared to p, the uncertainty

associated with the EB estimator is quite high. Thus, using the limited translation EB

estimator instead of the EB estimator does not cause much loss in Bai-; a risk. On the

other hand, when n is large compared to p and both n and p are large, much information

is lost by using the ML estimator instead of the EB estimator, and the limited translation

EB estimator is indeed a compromise between the ML and the EB estimators.

Another choice for the relevance function would be


pau) min(1, c/z~ ). i ,a 0 ifa> c~.

This relevance function reflects the idea that the EB rule is irrelevant for observations that

for which ||klS-E(Xi X,,)|| > co where co > c:, ansd the corresponding GR.LS is givenbhy


1-s,co E{1- p,co(aW)}2 = E{1 -min(1, cl\l/M)I[aW < c ]}2


= E [IaW c ] + {1 -~1 min(1, c/M/a)}I[aW < c ] 2





=P(W > c2/) 8 C2/a)E[W-lI{(c2 8) < < (C 8))

-(2c/ )E[W- I[(c2 8) < Il< C 8)~l









= P(Wo > c2, + (C2/a( l)p-17 C2/ 2 W < C a)

-(2/ )~p)I( )0- (p ) )P(c2/ 121 < c /aL), (4-19)

where Wi, i = 0, 1, 2, are same as in Equation 4-17.
4.5 Frequentist Risk of the Limited Translation EB Estimators

Our purpose in this section is to obtain an unbiased estimator of the risk of the
limited translation estimator. Recall that the limited translation estimator is given as

0 (" = Xi Br(X, X,,)pc(||klS- (Xa X,,)||2) and to shorten notation we write pc
for pc(||klS- (Xi X,,)||2). The riski of ac ("E is calculated as

Rl(ei, 0 ?" ) =E,(ei {(0 Of)(0i ~, ) }

= E a2CE~y S-1(Xi X,)(X; ,) Slp,2)

+t a~Ey { ( X4) (Xi X,)'S- pe) E

+t aIE~yS-1(X i X,)(04 X ) pc}. (4-20)

Using the multivariate version of Stein's identity, given in Lemma 4.2.2, we obtain the
following expression for the risk of the limited translation estimator.
Theorem, 4.5.1. TPhe frequetist risk of aLE ,,, can e exr d as


R1(0s, 0 (") = E + a2CE~y {S-(Xi X,,)(Xi X,) TSlp p2

+ 2aCEEyS'(Xi X,)(X, -X,) S pc

111



The proof is given in section C.4 of Appendix C.

Th~us, anl unbiased estima~tor of RI(84, 8 ~f) is given ans

171(0s, BL") = E + p 2ES- (Xi X,,)(X, X,,) S- E

+ 2ape [tr{S- (Xi X,)(X, ,)"} 1+ n;-1] ES E~





1 I[||kS 2 (Xi X, )||>c]
+ 2arpe (1- 1/n;)||S-(X, X,)||-2

xES- (Xi X,)(X, X,)TS-1E.


(4-22)


This quantity does not take large values even if Xi is far from X,. This is because of

the presence of the function pc in each of the terms that depend on Xi X,. When the

distance from Xi to X,, ats measulred. by ||klS-(Xi X,)||i, take~s on a large value,

bigger than c, pe = c/||kiS-T(Xi X,)|| takes on a small value. We are thus protected

from large frequentist risks, regardless of how well the assumed prior resembles the true

prior distribution.

4.6 A Simulation Study

We now undertake a simulation study to evaluate the performance of the proposed

estimators and compare them with the ML and EB estimators. Here we focus our

attention to the frequentist risk of the three estimators. Thus, in the first step of our

simulation study we fix values for the Os, i = 1,..., n. We take p, the dimension of the

vectors 04, to be p = 2 and the sample size to be n = 30.

In our first scenario, the Os are obtained by taking a sample of size n = 30 from the

contaminated model Os E (I E 0.9N2~(0, Al) + 0.1N2~(0, A2), Where


1.0 0.2 36.0 7.2
Al= 0.2 1.0 7.2 36.0 (


Note that the off diagonal elements of A2 WeTO So chosen to keep the correlations of 041

and 8i2 Same as those implied by Al, cor(0st, 8i2) = 0.2.

In the second scenario, we obtain the Os from the normal distribution Os


(2 ~~ ~ 45. N20,A3, hee 3 iS the variance-covariance matrix of the

contaminated normal distribution.

Figure 4-2 shows the Os obtained from the contaminated model as well as those
obtained from the normal distribution.


-23)









The second step of the simulation study consists of generating the Xi, i = 1,..., n.

For each of the two sets of Os, we generate the Xi firstly as X4|9 in 1 Iv2 i 1l) and

secondly as Xle|9in 2 Iv2 i, 2), Where



C1 I0.2 1.0 0.1.0 5.0 .


At the third step, we estimate each of the Os, i = 1,..., n, of the two sets. Using

the data that came from both fl and f2, We calculate the estimators Xi, Of" and

8f (". For the limited translation estimators, w-e consider three values for the constant, c.
Corresponding to generalized relative savings loss, 1 s, = .1 the value of c is c = 1.251,

while for 1 s, = .05 and 1 s, = .01 the values of c are 1.517 and 2.028 respectively.

The second and third steps are repeated R = 10, 000 times. Let 94 be any estimator

of Os. Then, th~e frequentist risk of Os, RI(0s, Os) = Eg(B Os)(e, Os)', is approxima~ted

by Ri(0s, Os) = R-l l,",(0, 0s,(, O,,)T, where Os.,, is th~e estim~a~te of 84 from? th~e

rth run. Since the R1 are matrices, we calculate their traces, ti = tr[R1(0s, Os)], and their

determinants, di = det [R1(0s, Os)], as one number summaries.

Now, for each of the two sets of Os and for each of the two samplings distributions,

fl and f2, We Summarize the distributions of the resulting ti and di, i = 1,...,30, by

reporting the minimum values (Qo), the 25th percentiles (Q0.25), the medians (Qo.so), the

75th percentiles (Qo.7s), the maximum values (Q1), the means (11. .is) and the standard

deviations (Stdev). The results are di;11l-phi I in Tables 4-1 4-4, where in each cell the

first entry describes the quantiles of the ti, while the second one describes the quantiles of

the di.

Some interesting issues emerge out of Table 4-1. First, even in the case where the

assumed prior is not very close to the true prior, the EB estimator performs well. As

far as the averages (11. ,1.) of ti and di are concerned, it does better than the MLE. On

average, the two measures of frequentist risk of the MLE are reduced by 1 1.843/1.998 =









7.'71.' and 1 .824/.958 = 13.9'*' respectively by the EB estimator. Actually, all the

entries of columns Qo Qo.7s of the row of Of" are smaller than the ones of the row of

Xi. However, the EB estimator can result in very large frequentist risks for observation

that are far fromX,, and this becomes obvious by observing the entries of Q1. The two

entries in Q~I for the estimator Of"~ are 3.'708/2.055 =1.804 and 2.584/1.015 = .546

times '?1 r-i than those of the minimax estimator, Xi. Comparing now the EB estimator

with the three limited translation estimators, we see that the entries in columns Qo Qo.75

are identical or almost identical. The limited translation estimators, however, have much

smaller maximum (Q1) risks than the EB estimator, since they become closer to the

minimax estimators for outlying observations. They thus have smaller average (jl \1..i) risk

than that of the usual EB estimator. For instance, Off,",, reduces the average risks of the

EB estimator by 1 1.758/1.843 = 4.61 and 1 .749/.824 = 9.10'; .

Similarly, in Table 4-2, we see that the EB estimator has maximum risks (Q1)

31.157/10.275 = 3.03 and 108.290/25.376 = 4.26 times bigger that those of the

MLE. However, the EB estimator reduces the average (11. ,1.) risks of the MLE by

1 7.084/9.989 = 29.0*' and 1 12.997/23.950 = 45.'7 :' respectively. The limited

translation estimators compare very favorably to the EB estimators. In particular,

th~e estimator Off,",, decreases the maximum? risks (Qi)j of th~e EB estimator by

1 12.473/31.157 = 59'11.'. and 1 35.406/108.290 = 67.t311'. respectively. It also

reduces the average (11. I1.) risks of the EB estimator by 1 6.319/7.084 = 10.H I' and

1 10.249/12.997 = 21.1 1' respectively.

Continuing to Table 4-3, which dli ph the results for the set of Os that was obtained

from the normal prior (2 and the fl sampling distribution, we see that the EB and the

limited translation estimators have very similar performance. The limited translation

estimators have slightly bigger Mean risks than the EB estimators but they slightly

decrease the maximum risks (Q1) of the EB estimators. Also, compared to the MLE, the

limited translation estimators, have smaller Mean risks but bigger maximum risks.











Table 4-4 di pE.--s~ the comparison of the estimators for the case where the Os

were obtained from the (2 prior, and the Xi from the f2 Sampling distribution. This

comparison is very similar to the one we have seen for the (2 prior and the fl sampling

distribution. Again, the EB and the limited translation estimators perform similarly.

Their slight differences are that the limited translation estimators have bigger Mean risks

than the EB estimators but they decrease the maximum risks (Q1) of the EB estimators.

Also, the limited translation estimators have smaller Mean risks but bigger maximum risks

than that of the MLE.

In order to study the effect of the sample size, we took samples of size 20 from each

of the two sets of thirty 94. Our new sets of Os are di;11l-p Id in Figure 4-3 (a) and (b).

The second and third steps of this study were same as the ones described earlier. The

results, shown in Tables 4-5-4-8, are very similar to the ones that we have already seen. It

is thus safe to conclude that the sample size does not affect much the performance of the

estimators under examination. We may note that for n = 20 the values of c corresponding

to 1 s, = 10I' 5' and 1 are c = 1.195, 1.443 and 1.910 respectively.




-- p=2, n=10
r- -- p=2, n=30
p=5, n=10
p=5, n=30








0 .0 5


Fiue4-.Poto -s sa ucio fcfrp= ,1 adn=O0 0






















O









*




-5* *




61















e e





~I
**o







-6 -4 -2 0 2 4

61


(b)

Figure 4-2. The 8; generated from (a) the contaminated model and (b) the normal model.














1.961 1.987 1.995 2.011 2.055 1.998 0.020
Xi
i0.925 0.948 0.956 0.967 1.015 0.958 0.020
EB 1.647 1.687 1.703 1.724 3.708 1.843 0.458
0f .65r7 0.690O 0.701 0.722 2.584 0.824 0.406i
gLB1.647 1.689 1.704 1.729 2.149 1.758 0.141
1.251,i 0.657 0.691 0.703 0.726 1.109 0.749 0.121

gLB1.647 1.687 1.703 1.725 2.218 1.761 0.158
1.517,i 0.657 0.6;90 0.701 0.722 1.176 0.752 0.136

gLB1.647 1.687 1.703 1.724 2.3863 1.772 0.194
2.028,i 0.657 0.6;90 0.701 0.722 1.335 0.761 0.169


9.807 9.935 9.975 10.054 10.275 9.989 0.102
Xi
i23.128 23.6389 23.889 24.1632 25.376 23.950 0.499
EB 4.766 5.086 5.227 5.420 31.157 7.084 5.949
5'B .588 6.302 6i.676 7.299! 108.2901 12.99!7 21.455
gLB4.953 5.3632 5.5632 5.840 12.473 6.319 2.016
1.251,i 6.021 6.981 7.522 8.4631 35.406 10.249 7.304

gLB4.8163 5.179 5.352 5.585 13.707 63.223 2.354
1.517,i 5.703 63.525 6.971 7.747 40.426 10.037 8.478

gLB4.768 5.092 5.238 5.4363 16.809 63.323 3.067
2.028,i 5.594 6.316 6.694 7.342 52.304 10.422 11.043


1.961 1.987 1.995 2.011 2.055 1.998 0.020
Xi
i0.925 0.948 0.956 0.967 1.015 0.958 0.020
EB1.525 1.612 1.704 1.805 2.577 1.767 0.228
e 0.564 0.63~2 0).706 0.79)3 1.322% 0.749 0.172
LB1.528 1.6334 1.7611 1.892 2.211 1.790 0.178
1.251,i 0.566 0.646 0.752 0.874 1.099 0.772 0.143
LB1.5263 1.617 1.722 1.871 2.322 1.776 0.196
1.517,ii 0.564 0.635 0.7211 0.845 1.173 0.760 0.156
LB1.525 1.613 1.704 1.814 2.534 1.769 0.226
2.028,ii 0.564 0.632 0.707 0.797 1.308 0.752 0.174


Table 4-1. Comparison of the risks of the estimators
the fl sampling distribution, a = 30.


under the contaminated model and


Q0.25


Q0.50


Q0.75


Mean


Stdev


Table 4-2. Comparison of the risks of the estimators
the f2 Sampling distribution, a = 30.


under the contaminated


model and


Q0.25


Q0.50


Q0.75


Mean


Stdev


Table 4-3. Comparison of the risks of the estimators under the normal model and the fl
samplingf distribution, a = 30.


Q0.25


Q0.50


Q0.75


Mean


Stdev

































9.831 9.941 9.994 10.081 10.236 10.013 0.098
Xi
i23.108 23.730 24.082 24.3631 25.150 24.089 0.474
EB 3.752 4.607 5.376 6.748 13.407 6.210 2.260
e 3.511 5.106 6i.703 9).910 2L4.874 8.6421 5.150
gLB4. 152 5.202 6.067 7.468 11.332 6.605 1.771
1.251,i 4.285 63.581 8.708 12.669 23.924 10.389 5.080

gLB3.903 4.874 5.731 7.213 12.041 63.412 1.984
1.517,i 3.794 5.749 7.689 11.515 24.5634 9.612 5.282

gLB3.767 4.647 5.434 63.882 13.019 63.255 2.207
2.028,i 3.540 5.188 6.870 10.298 25.045 8.888 5.307


Table 4-4. Comparison of the risks of
samplingf distribution, a =
Q o I 0.25 I


the estimators under the


normal model and the f2


Q0.50


Q0.75


Mean


Stdev
























O











e
e

0-I *, e



-8 -6 -4 -2 0 2

61


(a)







c**











S-6 *4 .2 .

*8

(b*

Figre -3 A ampe f B gnertedfrm () te ontmiate moeland(b)th nom*
0- *l














1.956 1.987 2.002 2.014 2.025 2.000 0.018
Xi
i0.922 0.945 0.96;0 0.970 0.992 0.959 0.017
EB1.646 1.665 1.6383 1.708 2.825 1.838 0.378
0f .659, 0.676i~ 0.689 0.710 1.691 0.8i22 0.3243
gLB1.647 1.672 1.6385 1.719 2.200 1.764 0.180
1.251,i 0.6;60 0.681 0.694 0.722 1.139 0.759 0.154

gLB1.646 1.665 1.6383 1.710 2.287 1.769 0.207
1.517,i 0.659 0.677 0.6;90 0.712 1.212 0.764 0.176

gLB1.646 1.665 1.6383 1.708 2.482 1.792 0.267
2.028,i 0.659 0.676 0.6;89 0.710 1.382 0.783 0.226;


9.834 9.908 10.041 10.087 10.256 10.012 0.112
Xi
i23.176 23.521 24.1163 24.234 25.035 24.009 0.515
EB41 1 ; 5.145 5.322 5.747 21.543 7. 137 4.509
5f .968j 6.461 6.824 7.846 691.8313 13.322 16.520
gLB5.2363 5.479 5.740 6.200 12.124 6.665 2.254
1.251,i 6.729 7. 365 7.996 9.196 31.778 11.490 8.317

gLB 5.050 5.271 5.492 5.948 13.079 6.596 2.623
1.517,i 63.245 6.797 7. 293 8.428 34.942 11.358 9.535

gLB4.954 5.157 5.340 5.773 14.8211 6.705 3.231
2.028,i 5.995 6.496 6.872 7.919 42.474 11.777 11.572


1.980 1.997 2.003 2.016 2.029 2.006 0.013
Xi
i0.945 0.956 0.965 0.976 0.986 0.965 0.011
S1.641 1.744 1.799 1.898 2.182 1.835 0.146
e 0.656 0.737 0.780 0.842 1.055 0.807 0.115
LB1.641 1.796 1.8611 1.974 2.125 1.871 0.140
1.251,i 0.656 0.764 0.837 0.905 1.036 0.840 0.115
LB1.641 1.763 1.825 1.939 2.180 1.8563 0.152
1.517,i 0.656 0.744 0.802 0.874 1.069 0.826 0.123
LB1.641 1.745 1.800 1.901 2.214 1.841 0.156
2.028,i 0.656 0.737 0.7811 0.844 1.082 0.812 0.124


Table 4-5. Comparison of the risk of the estimators under the contaminated model and the
fl sampling distribution, a = 20.


Q0.25


Q0.50


Q0.75


Mean


Stdev


Table 4-6. Comparison of the risk of the estimators under the contaminated model and the
f2 Sampling distribution, a = 20.


Q0.25


Q0.50


Q0.75


Mean


Stdev


Table 4-7. Comparison of the risk of the estimators under the normal model and the fl
samplingf distribution, a = 20.


Q0.25


Q0.50


Q0.75


Mean


Stdev

































9.861 9.920 9.940 9.988 10.057 9.958 0.053
Xi
i23.472 23.678 23.791 24.057 24.242 23.829 0.232
EB4.6385 5.964 63.588 7. 305 10.600 6.925 1.6338
e 5.45,5 8.235 10.149) 11.821 191.664 11.0'22 4.305
LB 5.006 6.517 7.233 7.842 10.205 7.342 1.430
1.251,i 6.216 9.791 12.6321 14. 170 20.892 12.760 4.412
LB4.798 63.242 6.947 7.641 10.451 7.164 1.5633
1.517,i 5.717 8.975 11.504 13.123 20.733 12.0111 4.555
LB4.694 6.012 6.668 7. 396 10.648 6.988 1.6354
2.028,i 5.475 8.354 10.416 12.182 20.162 11.271 4.478


Tabler 4-8 Cmparison of the risk


of the estimators under the normal model and the f2


samplingf distribution, a
Q o I 0.25


Q0.50


Q0.75


Mean


Stdev









CHAPTER 5
MULTIVARIATE EMPIRICAL BAYES AND LIMITED TRANSLATION
EMPIRICAL BAYES ESTIMATORS: THE CASE OF ALL UNKNOWN
PARAMETERS

In this chapter we develop estimators assuming that all the model parameters

are unknown. The regular EB and robust EB estimators are developed in section

5.1. The Bai-; a risk of the EB and limited translation EB estimators are evaluated in

sections 5.2 and 5.3 respectively. Sections 5.4 and 5.5 examine the frequentist risk of

the estimators. In section 5.6 we undertake a simulation study to further evaluate the

frequentist risk performance. Further, in section 5.7 we apply the empirical Ba-- a and

limited translation empirical BEi-;- estimators in order to estimate the average vitamin

intakes of HIV-negative drug abusers. The proofs of the results of these chapter are given

in Appendix D.

5.1 Development of Estimators

Here we develop empirical Bayes (EB) and some robust empirical BE-- a estimators,

namely the limited translation estimators, for the case where all parameters are unknown.

To this end, consider the following model


Xij|I e imi,(0s, E*),

Os Et NV,(pw, A), (5-1)


where j = 2,..., k and i = 2, ..., n. Let, E =k-1E* and Xi k-l CJ= X,,. In order

to derive Bai-;- estimators for the Os, note that for i = 1,2,..., n,


100in"i N,(as, c),

Os ;t NV,(pw, A). (5-2)


Thus, when all the parameters are known, the Bai-;- estimator of Os, i = 1, 2,. ., n, with

respect to the matrix loss function L1(0s, a) = (Of a)(Of a)T, is given by


xi? 1, E(A+E )- (A p)= (I B) i+ B= Xi B(X p),(53










where B = E(A + E)-]

The assumed model implies that marginally Xi N 1,(pw, A + E), i = 1, 2, .. ., n.

First, assume that A is known, but yw is unknown. The latter is estimated by X.=

(n"k)j-' CE X,- which is its M/LE.,I UMVUE~T and best equiva~riant estimator under

translation of the sample space. Replacing pw by X., results in the so called EB estimator,


Of" =(I, -B)X,+Bx.= 14- B(X,- X.), (5 4)


which is a weighted average of the MLE of Os, Xi, and the sample mean, X.. This

estimator, in contrast with the MLE, uses information included in the whole data set and

not just the data corresponding to the ith individual or population.

Additionally, we have defined the limited translation estimator of maximum

translation c as


ek" = X B(Xi- X.)
1 1-
xpc(||(1 -1/n) z (A + ) z(Xi .)||2), (5-5)


where pc(u) = min(1, c/ ~) is termed the relevance function. Its argument is the

standardized distance of Xi to X. and its purpose is to put a bound to the frequentist

risk of the EB estimator by controlling the amount of shrinkage of the MLE towards the

common mean, X..

Suppose now that E and A are also unknown. Let



i= 1
a k

i= 1 j= 1

The inverse of the unknown marginal variance-covariance matrix, (A + E)-l, is estimated

by aS-l and E*, the variance-covariance matrix of the sampling distribution, is estimated









by bV, where a and b are known constants. The resulting estimators are


c0 Xi = -B(X, X.)pc(||klS- (X, .|2,(5-8)

where B = abk-IVS-l and kl = af(1 1/76)-

In the following sections we compare the properties of these estimators. We begin

by considering their Bai-; a risk performance. In subsequent sections we also study their

frequentist risks by both deriving unbiased estimators of these risks and by undertaking a

simulation study. Finally, we apply the proposed inferential procedure in order to estimate

the long term average vitamin intakes of HIV-negative drug abusers.

5.2 Bayes Risk of the EB Estimators

We now investigate the Bai-; a risk performance of the EB estimators assuming the the

normal prior ( NV,(p,, A) is the true one. The following Theorem provides an expression

for the Bai-; a risk of the regular EB estimator.

Theorem 5j.2.1. Th~e B~r;, risk of 0,9"f, under the assumedl prior, is given by


rz(E, ~Of"j ~) =E{0 T )(6 O"

= + ab(k~ 1) [ab(n~ pJ 2)- {n6(k 1) +t 1} 2] BE

+ a2b2(n p 2)- (k 1)tr(B)E. (5-9)


The proof is given in section D.1 of Appendix D.

Consider now the quadratic loss function, L2, glVen by


L28i, a)=(i aT -1(04 a,). (5-10)


For fixed a = n p- 2, the choice of which results in an unbiased estimator of the marginal

variance-covariance matrix, as discussed in the previous chapter, the Bayes risk of the EB

estimator, under the L2 lOSS funCtiOn is minimized for b = {n(k 1) + p + 1}-1. With this









choice of a and b, the expression of the Bai-; a risk under L2 reduces to


T2(~ dEB) = E {(0, Of di)TE- (0, Of di)}= tr {E- r ((, Of ")}
(n -p -2)(k -1)
=p tr(B). (5-11)
n(k -1) +p +1

Clearly, r2(~ EB)i < 72 = a~i) that is, the EBR Ba;-r- cstimator haRs smaller Ba;-, risk
that the ML estimator.

The Bai;- risk can also be expressed as

k-1
r2(~ EBH)= Ta2 EB) + (n, p 2) {1 tr(B), (5-12)
n(k -1)+p+1

where Of" is the EB estimator of 8i assuming that E is known, that is, Of" = Xi

B( Xi X.) and B = ak-1E*S-l. Thus, the second term in the righthand side of

Equation 5-12 can be thought of as the price in terms of Bayes risk for having to estimate
E from the data and it converges to a matrix of zeros as either n or k increase.

5.3 Bayes Risk of the Limited Translation EB Estimators

We now obtain an expression for the Bayes risk of the robust estimator. This

expression is provided by the following Theorem.
Theorem 5.3.1. Let a, = n p 2 and b, = n-l(k 1)-1. Then, the B.r,a risk of the

limited translation estimator of maximum translation c is given by


rI(1(i, J"H) =E+a--2+a a )b 1)

+aa {12bb~ b2hlb (b I+1)}

+aa, b2n(k -1)(bl + 1)E{1 pc(aW)}2

+2bn(k ) {aa b(b; + 1)) Epc(aW)-J 1} BE

+a'b2 1 47-16; +f (k: 1)E~pc(aWI)2 1}tr(B)E (5 13)

where W ~ Beta((p + 2)/2, (n p 1)/2).

The proof is given in section D.2 of Appendix D.









For a = a, = n p 2 and b = {n(k 1) + p + 1}-1, and under the L2 lOSS funCtiOn,

th~e Ba;-; a risk of di (" is eqlua~l to


r2/ LCEB) = {0,O E ( (")}=tr{"rz> 0()
(n p-2)(k -1)
= p- {2E(pe) -Ep)t() (5-14)
n(k -1) + 1 U\/~~\

It is clear that the proposed estimator has smaller Bai-; a risk than the MLE. However, it

has slightly bigger Bai-; a risk than the regular EB estimator.

In order to quantify the last statement of the previous paragraph, we use the concept

of the relative savings loss of d (",f with respect to X,, defined as


RSV\VL( >1 4 r -0 lB- 2 E 2 zB] (5-15)

The term T2E xi) r72E sl0B) is the savings, in Bai-, risk sense, that occur when using
the EBa estimatorc, inta fte MLEnrrr~, while t2 dL1B) 72(~ zEB) is the loss that occurs

when- -- u ing 0 instead of the EB estimator. It can be shown, using Equations 5-11 and

5-14, thait R~SL,(er ("; X)- = 2E~(pe) + E~(p2) = E~(1 pe)2 Sc.
In Figure 5-1 we plot RLS against c and it can be seen that RLS is a decreasing

convex function of c. We can thus choose c by deciding by what percentage is worth

increasing the Bai-; a risk of the EB estimator in order to receive protection again large

frequentist risks. The constant c can be so chosen that RLS= 0.05 or 0.01. By sacrificing

5' or 1 of the Bai-; a risk, we receive considerable protection against large frequentist

risks. In order to make the latter point clear, we will examine and compare the frequentist

risks of Of" and 8f (" in the subsequent sections.

Finally, we give another expression for RLS


RLS = P(Wo > a- c2) C2( l)a)-1 2W > a-1C2)
1 p+1 p2 n+1 n
+tca-Hr()0( )r( )r- ( )P(W~1 > a-'c2), (5-16)
2 2 2 2

where Wi, i = 0, 1, 2, have the Beta((p + 2 i)/2, (n p 1)/2) distributions respectively.









5.4 Frequentist Risk of the EB Estimators

In this section we investigate the frequentist risk performance of the EB estimators.

We proceed by deriving unbiased estimators of their risks, a task which entails using the
multivariate Stein's identity which was provided in Lemma 4.2.2.

First, the frequentist risk is of EB calculated as


Rl(0s, 0f") =Eg [{0, +B(X, X.)}(B {O Xi +B(Xi .)} l], (5 17)

with the expectation being taken with respect to the sampling distribution of the Xij,

i = 1, n, j = 1, k, and assuming that a = { (01)", .. ,(0,)" } is fixed. Using
Stein's identity we obtain another expression for the risk.

Theorem 5.4.1. The ~freque~ntist risk: of diLEB can be expressed as


R,(0s, Of") =~ E 2b2n)(k 1)tr Ey { S- (X X, ) (X X.)"S- } EE

+abnr(k 1) [ab {n(k 1) + 1} + 2] ~E(yS-' (Xi X.)(Xi X.)TS-' }E

+ 2abn~(k 1)E~EB fr{iS-' ( X X.) ( X X.T) }S- E

-2abn(k 1)(1 1/n)~EE(S- )E.. (5 18)

The proof is given in section D.3 of Appendix D.
Let OjEB EB T EB T and conllsider ther losas obtained by averaging ther

individual L2 l0SSeS. The average quadratic loss, using the result of Equation 5-18, and for

a = n p 2 and b = {n(k 1)+ p + 1}-, can be shown to be equal to


R2EB: di'~=1-1) e Eg(0,Oi'-(0 f"
i= 1
(n 2) 2(k 1)
i= n- t1R(,E B)p n(k -1) +p +1

which for all 8 is less than the risk of the MLE. The latter is equal to R2(0, X) = p

and it is clearly N r;-i than the risk in Equation 5-19. In other words, the EB estimator

dominates the ML estimator, and thus it is a minimax estimator. An interesting approach









for proving the dominance of the EB estimator over the MLE is provided by Efron &
Morris (1972b).

Returning to the general result of Theorem 5.4.1, we can obtain an unbiased

estimator of the freqluentist risk of Ofj"f under the qlua~dra~tic loss L2. This estim~ator is

given by

1%2B, 8iL) sl B pn'b~n~k- (k -1)tr{S- (X, .)(X, X.)'IS- V}

+ab2nk- (k 1)b[ab~n(k 1) + 1} + 2]VS- (Xi X.)(Xi X.) S-1

+ 2abOnk-' (k ) tr {S-(X, X.)(X, X.)r} 1+ 1/nr VS- (5-20)

The above quantity will be large when Xi is far from X.. When k is large and the

risk associated with the ML estimators is small, observing an outlying Xi, might be an

indication that the corresponding 04 is far from the rest of the O's. Under this scenario, it

is not very wise to shrink by a lot the MLE of Os towards the center of the data. Intuition

as well as Equation 5 20 -11---- -i that, in such a case, the riski attached to Of" is quite

high.
5.5 Frequentist Risk of the Limited Translation EB Estimators

With? the purpose of showing that 0 ("R does n~ot allow h~igh? freqyuentist risks, we
obtain an unbiased estimator of its risk. To this end, recall that this estimator is given as

=jL~ X. -B(Xi. X.)pe, where pe pc(||klST(Xi X.)||2). F1TSt, the risk of (,~"
is calculated as




=Eege {0 xi +B(Xi X.)pc)(8 {O i +B(Xi X.)pc} (5-21)

Another expression of the above risk is given in the following Theorem.








Theorem 5.5.1. Th~e f~requen~tist risk of di (" can, be exrpressed as


Rl(Os, di(fi) = E
+a~2b2nb(k~ 1) {nhk 1~f) 1EE( S-1(X i X.)(Xi X.) Slp,2)

+a12b2n6(k 1)tr Eg S- (X, X.)(X, X.) S p }EEC (5-22)


+2abn(k 1)CEB tr{S'( ( X.) (X, X.) }S pJC
-2abnb(k 1)(1 1/n)CE~(S- pe)C

+2abn(k -1)CEy [S'(X, X.)(X, X.) ~S pc

Y(1 /ii)||S(Xi X.)||-2 E.*.E] (5-23)

The proof is deferred to the Appendix D, section D.4.
Based on the general result of Theorem 5.5.1, we obtain an unbiased estimator of the
risk of ij ?f under the quadratic loss L2. This estimator is given by

R2 8i ~L1B) )pn2b~nk- (k- 1)p tr {S- I(Xi X.)(X, X.)TS- V}

abil~n(k ) + 1}pe+ 2(1 1n)||S-E(i 1.)||2JXf1II
xab2nk- (k 1)pcVS- (Xi X.) (X, X.)TS-1

+2anb~rnk- (k )e t S X.)(X, X.j)}- 1 +1/n1 VS-' (5-24

Th~e above estimator of th~e risk -I I _-- -r that 8,r"f does not allow for large freqyuentist
risks. This is because of the presence of the function pc in each of the terms that depend
on Xi X.. When the distance from Xi to X. becomes larger than c, then pc
c/||kiS-(Xi X.)|| takes on a small vahlue not allowing the risk to become large.
5.6 A Simulation Study
We now undertake a simulation study to further evaluate the performance of the
EB and limited translation EB estimators. Since in previous sections we obtained closed









form formulas for the B .v. a risk of these estimators, here we focus our attention to their

frequentist risks.

Our simulation study here is very similar to the one of section 4.6. It consists of three

steps which we briefly describe. At the first step we obtain values for the Os, i = 1,. ., n,

and keep them fixed. We take n = 30 and the dimension of the Os to be p = 2. The

method of obtaining Os is exactly the same as the one described in section 4.6. We may

recall that the two sets of Os are shown in Figure 4-2.

At the second step of the simulation study we generate the observations Xij, i=

1,. ., n, j = 1,. ., k, where k was selected to be k = 3. For each of the two sets of Os,

we generate the Xij firstly as X,|9 in 1 -l rv2 i 1) and secondly as X,|9 in 2 f -

N2 (i, 2~), Where El and E2 are given in 4-24.

At the third step, we estimate each of the Os, i = 1,..., n, of the two sets using the

observations that came from? both? fl and f2. WeV calculate th~e estimators X,, 0,9" anld

Oj~fi. For th~e limited tranlsla~tion estimators, we consider three values for the con~stanlt
c. Corresponding to 1 s, = .1 the value of c is c = 1.251, while for 1 s, = .05 and

1 s, = .01 the values of c are 1.517 and 2.028 respectively.

The second and third steps are repeated R = 10, 000 times. Let Os be any estimator

of Os. Then, th~e frequentist risk of Osi, Ri(Os, Os) = FEg (0, Os)(0, Os)7}, is

alpproxim~ated by i,(0s, Os) = R-' C," (0 0,,)(O, Os,,)", where Os.,, is th~e

estimate of 8i from the Tth run. Since the171 are matrices, we calculate their traces,

ti = tr [R1(Bi, Os)], and their determinants, di = det [R1(Bi, Os)], as one number summaries.

For each of the two sets of Os and for each of the two samplings distributions, fl

and f2, We Summarize the distributions of the resulting ti and di, i = 1,...,30, by

reporting the minimum values (Qo), the 25th percentiles (Q0.25), the medians (Qo.so), the

75th percentiles (Q0.75), the maximum values (Q1), the means (11. .is) and the standard

deviations (Stdev). The results are di;11l-p Id in Tables 5-1-5-4, where in each cell the two

entries describe the quantiles of the ti and di respectively.









Observing Table 5-1, we first notice that the entries of the row that corresponds to

the MLE are very close to the theoretical values, tr(k-1Ex) = 0.667 and det(k-1Ex)=

0.107, with very small variability. Now, even in the case where the assumed prior is not

very close to the true prior, the EB estimator performs well. The averages (11. ,1.) of

ti and di for the EB estimator are smaller than those of the MLE. On average, the two

measures of frequentist risk of the MLE are reduced by 1 0.649/0.666 = 2.55' and

1 0.101/0.107 = 5'1 !'. respectively by the EB estimator. Not only the means but

a~ll the en~tries of columnls Qo G.75 corresp~on~ding to ("f a~re smaller thanl the ones

corresponding to Xi. The bad property of the EB estimator is that it can result in large

frequentist risks for observation that are far from X.. This becomes obvious by observing

the entries of QI. The two entries in QI for the estimator diE ar~e 0.863/0.6j80= 1.27

and 0.173/0.110 = 1.57 times bigger than those of the minimax estimator, Xi. We

now compare the three limited translation estimators to the EB estimator. Firstly, we

observe that the entries in columns Qo Qo.7s are identical. The limited translation

estimators, however, have smaller maximum (Q1) risks than the EB estimator, since

they become closer to the minimax estimators for outlying observations. They thus have

smaller average (11. ,ii) risk than that of the usual EB estimator. For instance, 8~L~

reduces the maximum risks of the EB estimator by 1 0.686/0.863 = 22.05' and

1 0.113/0.173 = 34.1.*'

In Table 5-2, we see that the EB estimator has maximum risks (Q1) 7.282/3.332=

2.19 and 8.953/2.780 = 3.22 times '?i1 ;-- that those of the MLE. However, it reduces the

average (11. ,is) risks of the MLE by 1 2.938/3.332 = 11.52'. and 1 2.127/2.666 =

20.2'O' respectively. The limited translation estimators compare very favorably to the EB

estimators. In particular, th~e estimator ijfr,4 dcrea~ses th~e maximuml~ risks (Qi1) of the

EB estimator by 1 3.746/7.282 = 48. .' and 1 3.300/8.953 = 63.1 1' respectively. It

also reduces the average (11. I1.) risks of the EB estimator by 1 2.752/2.938 = 6.3 :' and

1 1.852/2.127 = 12.CI' respectively.










Tables 5-3 and 5-4 display the comparison of the estimators, for the case where

the Os were obtained from the normal prior (2, and the Xi from the fl and f2 Sampling

distributions respectively. In both Tables we see that the EB and the limited translation

estimators perform very similarly. Their slight differences are that the limited translation

estimators have bigger average risks (11. ,1!) than the EB estimators but they decrease

the maximum risks (Q1) of the EB estimators. Further, the limited translation estimators

have smaller Mean risks but '? r-i maximum risks than that of the MLE.

In order to study the effect of the sample size, we took samples of size 20 from each

of the two sets of thirty 94 that were obtained in the first step of the simulation study, see

Figure 4-3 (a) and (b). The second and third steps of this study were same as the ones

described earlier. Note that for n = 20, the values of c that correspond to RLS= 101' 5' .

and 1 are c = 1.195, 1.443 and 1.910 respectively. The results were very similar to the

ones that we have already seen.

5.7 Application

In this section we apply the proposed inferential procedure in order to estimate the

'longf term average' vitamin intakes of HIV-negfative drug abusers. In this application, we

will be using the baseline data from a prospective cohort study of the role of drug abuse in

HIV/AIDS weight loss and malnutrition conducted in Boston, Massachusetts, USA.

Each of the n = 54 subjects completed 3-day- food records, recording type and amount

of food, including supplements and vitamins. Dietary analysis was performed on the 3-d~i-

food records and daily nutrient intake was determined. The intakes of several nutrients

were determined but here, for the sake if simplicity, selected for analysis only two of those

nutrients, specifically, vitamin A and Thiamin (also known as vitamin B1).

The observed distribution of the intakes of the two vitamins is not close to a

realization from a bivariate normal distribution. This indicates the need of transforming

the data before applying methods that require normal distributions. We thus start our










analysis by considering a bivariate Box-Cox, Box & Cox (1964), transformation. It turns

out that the values of As, i = 1, 2, for the transformation are At = 0.10 and X2 = 0.35.

The average intakes of the two vitamins of the n = 54 subjects, after the transformation,

are di11l li-k & in Figure 5-2, and it is clear that even after the transformation the

assumptions of normality are not exactly met. For this reason, a robust procedure,

like the limited translation estimators, would be more appropriate than the regular EB

estimators.

Let Xijl denote the intake of vitamin A of person i in d~i- ), and, likewise, X ._. the

intake of vitamin B1 of person i in ami- Further, Xij = (Xijl,X ..)T is the response

vector of person i in d~i- ). Additionally, 8il and 8i2 denote the average daily intake of

vitamin A and B1, respectively, of person i. The vector Os = (041, Bi2)T is aCCOrdingly

defined. Now, the EB and limited translation EB estimators are derived based on the

assumed model


xij~ I in N2 1V i, C

8 id N2 1(p, A), i = 2, .. ,54, j= 1, 2, 3, (5-25)


which can equivalently be written as a mixed linear model since,


X = pw + Os + Eij

8i d 12 (0, A), Eij d 1V2 (0, E), i = 1, 2, 54, j = 1, 2, 3. (5-26)


Table 5.7 di pl .ss~ the estimated long term average intakes of the two vitamins for

the first ten patients in our sample. The estimates were obtained using the ML, EB and

limited translation EB estimators. The first column under each heading refers to the

estimated vitamin A intakes while the second one refers to the intakes of vitamin B1.

Note that the average vitamin A and B1 intakes, after transformation, are X.=

(9.096, 0.721)' while (n 1)-1S, the estimated marginal variance-covariance matrix is



























































I I I I I I I


given by (n 1)-1S = .121 01.6 Also note that for the limited translation estim ator


we have chosen c = 1.301 which corresponds to a relative savings loss of 1 .s,. = 0.10.

Fr-om Table 5.7, we see that the EB estimator pools the ML estimates towards the

grand mean. For those ML estimates that are close to the grand mean, the EB and the

limited translation EB estimates are identical while for those that are far front the grand

mean, the limited translation estimates are somewhere between the ML and EB estimates.

Similarly, in Figure 5-3, we see the ML, EB and limited translation EB estimates for

all n = 54 subjects.


p=2, n=10
p=2, n=30
p=5, n=10
p=5, n=30


0.0 0.5 1.0 1.5
c


2.0 2.5 3.0


p,.(aW)}2 as function of c, for p = 2, 5 and n = 10, 30.


Figure 5-1. Plot of RLS= E {1














0.654 0.664 0.666 0.669 0.680 0.666 0.006
Xi
i0.103 0.106 0.106 0.108 0.110 0.107 0.002
EB0.617 0.629 0.634 0.641 0.863 0.649 0.055
0.092 0.095 0.097 0.099 0.173 0.101 0.017
LB0.617 0.629 0.634 0.641 0.686 0.638 0.017
1.251,i 0.092 0.095 0.097 0.099 0.113 0.098 0.005
LB0.617 0.629 0.634 0.641 0.695 0.639 0.019
1.517,i 0.092 0.095 0.097 0.099 0.115 0.098 0.006;
LB0.617 0.629 0.634 0.641 0.716 0.640 0.023
2.028,i 0.092 0.095 0.097 0.099 0.121 0.099 0.007


3.280 3.314 3.328 3.353 3.401 3.332 0.030
Xi
i2.584 2.6332 2.659 2.699 2.780 2.666 0.048
2.542 2.5638 2.610 2.6352 7.282 2.938 1.073
e 1.566 1.6i02 1.660 1.7214 8.9583 2.127 1.5831
LB2.5463 2.577 2.6322 2.680 3.746 2.752 0.336
1.251,i 1.574 1.610 1.679 1.764 3.300 1.852 0.469
LB2.543 2.570 2.6312 2.656 3.926 2.753 0.380
1.517,i 1.567 1.603 1.664 1.729 3.5633 1.855 0.531
LB2.542 2.5638 2.610 2.6352 4.367 2.778 0.474
2.028,i 1.566 1.602 1.660 1.724 4.200 1.889 0.66;9


0.646 0.660 0.666 0.672 0.690 0.666 0.009
Xi
i0.101 0.104 0.106 0.108 0.114 0.106 0.003
EB0.592 0.619 0.632 0.650 0.745 0.640 0.034
0~ .085 0.09)2 0).09!6 0.10)2 0.1215 0l.098 0.009)
LEB, 0.592 0.621 0.636 0.666 0.696 0.643 0.029
1.251,i 0.085 0.093 0.097 0.106 0.116 0.099 0.008
LB0.592 0.619 0.633 0.659 0.708 0.641 0.030
1.s17,i 0.085 0.092 0.096 0.104 0.118 0.099 0.009
LB0.592 0.619 0.632 0.651 0.740 0.640 0.034
2.028,i 0.085 0.092 0.096 0.102 0.124 0.099 0.010


Table 5-1. Comparison of the risks of the estimators under the contaminated model and
the fl sampling distribution, a = 30.


Q0.25 I 0.50 I 0.75


Mean


Stdev


Table 5-2. Comparison of the risks of the estimators under the contaminated model and
the f2 Sampling distribution, a = 30.


Q0.25 I 0.50 I 0.75


Mean


Stdev


Table 5-3. Comparison of the risks of the estimators under the normal model and the fl
samplingf distribution, a = 30.


Q0.25 I 0.50 I 0.75


Mean


Stdev














3.233 3.314 3.329 3.358 3.418 3.338 0.040
Xi
i2.518 2.637 2.661 2.706 2.806; 2.674 0.061
EB2.213 2.406 2.6328 2.902 4.458 2.772 0.506
1.205 1.412 1.668 1.992 3.721 1.838 0.594
LB2.232 2.4711 2.758 3.075 3.779 2.832 0.403
1.251,i 1.223 1.486 1.827 2.264 3.099 1.933 0.507
LB2.217 2.425 2.6382 3.013 4.004 2.800 0.445
1.517,i 1.208 1.433 1.732 2.165 3.328 1.887 0.551
LB2.213 2.407 2.6332 2.919 4.375 2.779 0.501
2.028,i 1.205 1.413 1.672 2.022 3.678 1.852 0.601


0.653 0.660 0.666 0.671 0.678 0.666 0.007
Xi
i0.102 0.105 0.106 0.108 0.109 0.106 0.002
EB0.618 0.626 0.633 0.640 0.750 0.647 0.042
e 0.0938 0.095j 0).096 0.099! 0.133 01.101 0.012
LB0.618 0.626 0.633 0.641 0.682 0.638 0.018
1.251,i 0.093 0.095 0.096 0.099 0.111 0.098 0.005
LB0.618 0.626 0.633 0.640 0.693 0.638 0.021
1.s17,i 0.093 0.095 0.096 0.099 0.114 0.098 0.006
LB0.618 0.626 0.633 0.640 0.718 0.641 0.028
2.028,i 0.093 0.095 0.096; 0.099 0.120 0.099 0.008


3.244 3.309 3.324 3.3463 3.374 3.322 0.032
Xi
i2.527 2.6328 2.656 2.6384 2.722 2.6348 0.048
EB2.475 2.547 2.575 2.680 5.275 2.928 0.837
,i!! .499, 1.594 1.627 1.744 5.606, 2.19119
LB2.487 2.559 2.599 2.725 3.759 2.781 0.402
1.251,i 1.512 1.610 1.6353 1.807 3.2863 1.904 0.558
LB2.477 2.549 2.580 2.693 3.951 2.787 0.465
1.517,i 1.501 1.596 1.6332 1.762 3.557 1.915 0.645
LB2.475 2.547 2.575 2.6817 4.388 2.831 0.590
2.028,i 1.499 1.594 1.627 1.745 4.1635 1.977 0.819


Table 5-4. Comparison of the risks of the estimators under the normal model and the f2
samplingf distribution, a = 30.


Q0.25 I 0.50 I 0.75


Mean


Stdev


Table 5-5. Comparison of the risk of the estimators under the contaminated model and the
fl sampling distribution, a = 20.


Q0.25 I 0.50 I 0.75


Mean


Stdev


Table 5-6. Comparison of the risk of the estimators under the contaminated model and the
f2 Sampling distribution, a = 20.


Q0.25 I 0.50


Q0.75


Mean


Stdev



















0.655 0.661 0.6;63 0.665 0.680 0.665 0.006;
Xi
i0.103 0.105 0.106 0.107 0.111 0.106 0.002
g 0.622 0.633 0.643 0.653 0.683 0.646 0.020
S0.094 0.096; 0.100 0.102 0.112 0.100 0.006;
LEB"f 0.622 0.636 0.653 0.662 0.694 0.651 0.020
1.251,i 0.094 0.097 0.102 0.105 0.113 0.102 0.006;

LEB"f 0.622 0.633 0.646 0.657 0.695 0.648 0.021
1.s17,i 0.094 0.096; 0.100 0.103 0.113 0.101 0.006;

LEB"f 0.622 0.633 0.643 0.654 0.6;88 0.647 0.021
2.028,i 0.094 0.096; 0.100 0.102 0.114 0.101 0.006;


3.249 3.302 3.324 3.349 3.380 3.327 0.035
Xi
i2.530 2.6323 2.662 2.694 2.758 2.660 0.056

ELB 2.460 2.747 2.850 3.015 3.66b8 2.917 0.321
1.477 1.794 2.001 2.147 2.884 2.043 0.401
LB2.469 2.838 2.989 3.1563 3.5632 2.995 0.301
1.251,i 1.487 1.900 2.205 2.339 2.845 2.157 0.394
LB2.4631 2.782 2.918 3.096 3.660 2.964 0.327
1.517,i 1.478 1.833 2.103 2.252 2.932 2.109 0.421

LEB" 2.460 2.750 2.856i 3.025 3.714 2.930 0.337
2.028,i 1.477 1.797 2.010 2.160 2.951 2.060 0.425


Table 5-7. Comparison of the risk of the estimators under the normal model and the fl
samplingf distribution, a = 20.


Q0.25 I 0.50 I 0.75


Mean


Stdev


Table 5-8. Comparison of the risk of the estimators under the normal model and the f2
sampling distribution, a = 20.


Q0.25 I 0.50 I 0.75


Mean


Stdev








































5 6 7 8 9 10 11 12
Vitamin A


Table 5-9. MLE, EB, LT Estimates
MLE EB LT
9.143 -0.171 8.451 0.086 8.760 -0.029
10.415 1.059 9.973 1.017 9.973 1.017
10.774 1.215 10.260 1.143 10.260 1.143
8.134 0.759 8.670 0.708 8.670 0.708
9.798 1.038 9.665 0.976 9.665 0.976
5.992 -0.521 6.700 -0.295 6.395 -0.392
11.537 2.178 11.340 15 11.438 2.018
9.319 1.430 9.732 1.236 9.638 1.280
10.412 1.054 9.968 1.013 9.968 1.013
8.890 0.518 8.847 0.567 8.847 0.567


g


W
* *
So








Figure 5-2. Average intakes of vitamins A and B1.



















































O e T



wer *
Att

m eA
O- e

In Vitamin A


Fiue5-.Etiae itks fvtaisA n o










CHAPTER 6
SITAINARY AND CONCLUSIONS

In this dissertation, we developed some robust estiniators of the multivariate normal

mean, namely the limited translation B la-< a and empirical B 11-c 4 estiniators, by extending

the work of Efron & Morris (1971, 1972a).

The multivariate limited translation B is-c < estiniators serve as a compromise

between the B la-< a and the nmaxiniun likelihood estiniators. We have demonstrated

the usefulness of such estiniators over the usual B is-c estiniators, in a B is-c risk sense,

under nmisspecified priors. Front the criteria of frequentist risks, we have demonstrated the

usefulness of such estiniators, when they are used for estimating parameters which depart

widely front the assumed prior means.

Additionally, we developed multivariate limited translation empirical B 11-c estiniators

of the normal mean vector which serve as a compromise between the empirical B 11-c

estiniators and the nmaxiniun likelihood estiniators. We examined the properties of such

estiniators and demonstrated their usefulness front the frequentist risks criteria, when

there is wide departure of an individual observation front the grand average.

Future work will develop similar estiniators using, however, shrinkage estiniators, as

those -II_a---- II .1 Efron & Morris (1976) for the unknown variance-covariance matrices.

In the process, we will address directly estimation of the variance-covariance matrix of a

multivariate normal distribution in a very general set up.









APPENDIX A
PROOF OF THEOREM 2 SERIES

A.1 Two General Results

We first prove two basic lemmas useful to the proof of Theorems 2.4.1 and 2.5.1.

Lemma A.1.1. Let Y ~ NV,(rl, aE) where a > 0. Then, for r: fit.;;ed scalars b and d and

i t.;, Red p-dimensional vector 4, we have


E{ I[||(aE)-E(Y ~)||2 < b]}
||(aE) 2 (Y 4)||2d



Proof. Wlie write Q = a-(Y ~) E- (Y 4) and observe that Q ~ y (A) where

A = (rl ~) (aE)-l(rl ~)/2. In what follows we repeatedly use the result that
if X ~ X(a),~ then the~ density, function of ,;,~;,,, X sa ifnt smo 2kVrals

k = 0, 1, 2,. ., with Poisson weights (e- Ak) /(k!).

We begin with the equality


E{||(aE)- (Y ~)||-2dllaC-( 12 < b]}


(A-2)


We now differentiate both sides of Equation A-2 with respect to rl. First note that


(Y rl) (aE)- (Y rl) = 2(aE)- (rl Y).




dY

(aE)- E{ I[||(aE) I(Y ~)||2 < b]}
||(aE)-a(Y ~)|| 2d

)(k!)- E Fx( ,+2 -d JX+2k < b]} (e^ k
k=0l


(A-3)










(A-4)


Hence ,









Now,


8(e-x Ak


e-' Ak e- kX'k-1


(C-1 9l ~)e-X k-1(k


A). (A-5)


Combining Equations A-4 and A-5 we obtain


||a )TY )||
G( ) e -(k


Ir[||(aE)-a(Y


A)E{(+2 -~d +2k < b]}


Ak'
~1E{(X 2+2k -dlX +2 2k: < b]}


(17 ~) [Ex { t9()"[ 2 -d+2(A) < b]}


(A-6)


This completes the proof of Lemma A.1.1.

Lemma A.1.2. Consider the same setting as in Lemma A.1.1. Then


||(aE)-a(Y -


- 4)T
I[||I(aE)-E(Y
~) ||2d


(4-)(-) Ej() ) {[7 ([i(A)]- I[y(A)
-2E { [S 2 -d +2X,(A) < b]}


+t aCEE {[ 2 X-drX +2(A) < b]}.


(A-7)


Proof. We start by differentiating twice both sides of Equation A-2 with respect to rl.
Note that


82
exp{


4) (aE)- (Y 4)/2}


(aE) 1(Y rl)(Y rl) (aE)- exp{-(Y

(aE)- exp{-(Y rl) (aE)- (Y 17)/2}.


rl) (aE)- (Y


r1)/2}


(A-8)


~)||2 < b]}


e
(4 )
k=0


E { [) (A)- "I [X.(A) < b]} .


~)||2 < bl}









Thus ,


(Y l) (Y
(aE)-1E{
||I(aE)- (Y


(A-9)


where


82 6-xXA k
{(aE)-l(r
84847l 847T
(aE)-le- Akl~k-1 X


~)e Ak-1 .k


(aE)-lr _)r )T(aE)-le- Ak-2 X2 + k(k

Substitutingf the last expression in Equation A-9 we obtain


1) 2kA}.


~(k!)- E{(X(Z*d 2k +2k< b]} i2(e Xkdrrl


,-A k-2
X e 2X + k(k 1) -2kA}E{((X 2k~d XJ2k < b]})


00-A k-1
+(aE)-1' k


- )E{((k 2k 2k < b]}k


E {[t)(A)] Ilk;i(A) < b]} (A-11)


Combiningf Equations A-9 and A-11 and collecting terms we obtain Lemma A.1.2.

Remark The results of Lemmas A.1.1 and A.1.2 hold even if we change the inequalities
from < b to > b with obvious modifications.


- 4)T
I[||(aE) 5(Y ~)||2 < b]}(aE)-
~) ||2d


(aE)-1E{||(aE) ~(Y ~)||-2d ~l~C-~( )2 < b]}
(k!) E{(;+2k+2k b]}2 Ak ,
k=0rT


(A-10)


x [Ex {[k(X) ":(A)< b]}+ {[\;I+4(" +4t~(A)< b]}

-2E( {[\+2(X dl+2(A) < b ]}


+ (aE)-1 Ex {[ 2 J"I\ +2(A) < b]}









A.2 Proof of Theorem 2.4.1

Proof. Let QI= (X pU)TE- (X p) anld observe that Ql|9 ~ X (A), where

A = (8 pW)TE- (8 p)/2. We thus have the following equality


/ e-E (X0) 2 dX
[I (X-p)l a29(i+g)] ||E-(X p)||2ad(2x)i2|E|2

=e~" lQ E{ C(1 +2 -d +k 2(+g)} (-2


which is the same as Equation A-2 with

a = 1, b = c2 (1 + g), rI = 0 and = pw. (A-13)

Since Equation A-2 was the basis of proving Lemmas A.1.1 and A.1.2, the results of these

two Lemmas hold for the special case defined by the Equations in A-13. This gives us the

following two qualities

X-0
E{ I[||E-a(X pw)|| 2 C2(1+gl
||E- (X pw)|| 2d

=(e p) E( { [ 2 ~(-d 7,2 2X C2(1+gl
-E {x( )] I- y()< 21 g] (A14)


and

(X 8) (X 8)T
E{ I[||E- (X pw)|| 2 C2(1+gl
||E-z(X pw)|| 2d
=(6- p)(6- p)' E[y (A) I I (A)
+E{[@p+4 -d C+4 X) C2(1+ g)]} 2E{[ Cp2 -d C+2 C

YU LC+ CEE [7 +2 )-dl i +2 X) < C2 (1 + g)] }. (A 15)

Also, note that for any k

11
pc,, (||(1+g)-EE (X r)||2> k Pc(l+-1 1)









I[Q1 < c2 1 + )l C 1~( + g)Q~ 1 > C2(1 + g)].


(A-16)


We now write


R1(0, Of ")


~Eg(Of~ e)(Of" ej)}


Ee [{X (1+$ g)- (X
x {X (1 + g)-1(X -


- p)pc{(1+ g) Qi}

p)pc{(1+ g)- Q } -


- 0}

0}'


Eg {(X B)(X B)T}


(1 + g)-1Ey [{(X


e)(X U)T + (X p)(X 8)7}pc{(1+ g)- Q }]


+(1 + g)-2Eg[(X p)(X p) p {((1 + g)-1QI}].

Because of the form of the function pc(.), we need to calculate


(A-17)


Ep7 [{(X B)(X pu)T + (X


Il)(X 8) }I[Q1 < c2( tgl


~E [{2(X-0)(X


-0) + (X


e)(e p)


+t(e p)(X 8)7}I[Q1 < c2(1 + g)l] ,


(A-18)


and


Eg {(X p)(X p)7I[Q, < c2(1+~/1

Ee [{ (X 8) (X 8)' + (X 8) (8 CL)T (6


p) (X 8)T


+(e cl)(e pU)T}I[QI < c2(1+ g)1].

We thus apply Equation A-14 with d = 0 to obtain

~E {(X )I[QS < c72(11g


(A-19)


(e V)(PIX f+2() (I g)C2


PX()( 9)1 g)C2}


(A-20)


and Equation A-15, again with d = 0, to obtain

~E( (X e)(X 8) I[Q~ < c2( yl








= 1 L/pIP[Xt2X 2 1 g)C2 ( _W( rUT

XC (P +~~4(X I1 g)C2 1 g)C2]
-2P[ +2() (1 9)C2] (A-21)

Furthermore, the form of the function pc(.), given in Equation A-16, and Equation A-17
indicate that we also need to find an expression for

G LEe [{(X 8)(X L)T + (X p)(X e) }Ql~ I[Qx > c2'l+sl

=2Ey {(X 8)(X B) Q 'I[QI > c2( gl

+tEe [{(X 8)(8 p)7 + (8 p)(X 0))}Q IZ[QI > c2(1+f g)l] (A-22)

The results of Equations A-14 and A-15 with d = 1/2 and reversed inequalities show that

G= 2(8 ) (e )r E { [S +4 4 ~I\~+( ) > C2( gl
Ex { [72 2 > C2

+2~EE{[y%,X 1i +2 2 () > C2(1f g)]} (A 23)

Finally, we rewrite

Eg {(X p)(X p) Q Il[Q, > c2(1~1

=Eg {(X B)(X 8) Q Il[Q1 > c2(1+gl

+t Ee [{(X 8)(8 p)T + (8 p)(X 8))}Q I[QI > c2(1+gl

+t E8 ((6 pu)(8 p) Q I[Q, > c2(1 +t g)]}. (A-24)

Again the results of Equations A-14 and A-15 with d = 1 and reversed inequalities show
that


Eg {(X p)l(X p) Q I[Q1 > c2(1+gl

=(e 8) (8 Ip) E { [X +4 X 1 [i+4( > C2(1+gl

+ cEE {[7+2(X -1 x+2 x> C12(1+1 g)]}. (A-25~)









Combining Equations A-16-A-25 and collecting the coefficients of E and (8 w) (8 w)T

separately, the result follows. O

A.3 Proof of Corollary 2.4.2

Proof. First, we obtain an expression for R2 8, 8LB) by directly using the result of
Theorem 2.4.1. That is, we calculate R28 8LB) __ t-1 1(B 8 LB)]. The resulting

expression can be simplified by making use of the two qualities that follow. First,


pE { [: +2 a()-1li 2(X > C2( )1

SAXE { [U7 +4)-[ -1 4( > C2(1~1

=XX P(- p +2k +2k > (1 g)C2
k.O 20

+ 2 p+ +'I2k +22k > (1 g)C2]
I- kx :p+4a p+2k
= -X P(- p+ +2k +2k > (1 g)C2
k.O 20
k -X ~ 0 2""
+ 2 p+aa~ +2 ?.+2k > ( 1g)C21
k.O 20

p +2k (p 2k)P[X@ 2k > (1 g)C2]
k=0 2
= X (A) > C2(1 g)], (A-26)

and similarly


pE { [7 +2 +2)1i [?~I()>C2( g

S AXE { [U7 4 +4 r~:,( > C2(l~/ 1

I' Ch P( ~ ,222k 1+2 > (1+ )C2]
kO 2 spl~
6-A k Z p+3+2k
+ 2 24+2k P+3+2k )C

,-xX k p+1+2k)
=~ PI 222+12k > (1 + )C2]
k=0 2+2









e- Akk F(p+1+2k oa ~ 2

6-A k p+1+2kip12
k 0 22+2 (p + 2k)P[ > 2k> 1g)C
k=O 2
6-A k p+1+2k -1
k= k (p~r ) ai+9, 0( ~~)2 z

=E {[yf,(A)]l Iy,(A)> c2(1 + g)]}.


(A-27)


The proof of Corollary 2.4.2 is now complete.


A.4 Proof of Theorem 2.5.1

Proof. Write B* = (1 + g*)-l, B = (1 + g)-l and Q2 1 BQ
Note that under the N,,(pw*, *) prior, the posterior is


B*(X pW)TE- (X


0| ~ N(O =7:8 (1 B*)X + B*p*, (1 -- B*)E).


(A-28)


Also, marginally X ~ NV,(pw*, (1 + g*)E) so that Q2 ~X, Where A = B*(pw*
rW)TE- (p* pU)/2. The risk under prior (* of the limited translation estimator is
calculated as


rl(E*, Of") =E{((

E{(( Ofp + Of! -

rz(E*, Of) +~ E{(Of


- Of")(e Of ") }

Of")(e Of + Of Oft") }>

- OF")(O~f OF")'},


(A-29)


with the last equality following from the fact that E{(( O )(Of OfB)"}= 0. We also

have that r z((*, Of) = (1 B*) E, and we need only to calculate E {( Of OfB) ( Of -

Of"B)T}. Note that Of Of"B = -B*(X pU*) + B(X rw)pc(BQ1). Hence,


E{(Of Of"B)(Of OfL")"}

(B*)2E{(X pw*)(X pw*)T} + B2E{(X -

BB*E[{(X pw*)(X pw)T + (X pw)(X


rW)(X pW) p (BQ1)}

- p*)T}pc(BQ )].


(A-30)









Rewritingf


(X w) (X pW*)T = (X pU*) (X -- p*)T + (pU* -- U) (X -- p*)"


(A-31)


and


p) (X pW)T = (X pU*)(X

pw*)(pW* pU)T + (pU* -- U) (X


pw*) + (pW* rU) (p* -- U)T

pw*) ,


(X

+(X


(A-32)


it follows from Equation A-30 and some simplification

E{( ( -~ fiF"(e e- ) }= B2 a s _

+ E{(X pw*)(X pW*) [B* Bp 2(BQ1)]2)

-~ BE (X *(p )T + (p* p)(X *)T}

x p,(clI(B){* pc(BQ I) } .


~WTE~p (BQ )}


(A-33)


For any k > 0, we write


E{(X pw*)(pw* pW) p (BQ )}

= E{(X p*)(w p) I[B*Q, Ic2B*B-11

+Ck a B*- E{(B*Q1) -(X pw*)(pw* pw) I[B*Q1 > c2 pa -1]},


(A-34)


and the two components of this expectations are calculated by applying Lemma A.1.2 and
the remark following it with Y = X, a = 1 + g*, b = c2( + g( *-1,r w* w
and d = 0 and k/2 respectively.
We also have that


E{(X pw*)p (BQ1)} = E{(X pw*)I[B*Q1 < c2 pa -11

+ck a B*- E{(B*Q1)- (X pw*)I[B*Q1 > c2 pa -1]}.


(A-35)










Here, the two components of this expectations are calculated by applying Lemma A.1.1

and the remark following it with Y = X, a = 1 + g*, b = c2( + g( *-1, rl= a*

S= pw, and d = 0 and k/2 respectively.

Finally, we collecting the coefficients (pw* pw) (p* pw)T and E separately and the

result follows. O

A.5 Proof of Corollary 2.5.2

Proof. We first calculate T2~* 8,LB) __ t(-171~* \ LB)} and we simplify the resulting

expression using Equations A-26 and A-27. O









APPENDIX B
PROOF OF THEOREM 3 SERIES
B.1 Proof of Theorem 3.4.1

Proof. Let Q = (Xi X,)TE- (Xi ,) and recall that B = (1 + g)-l. Also, recall
that Xi X, ~ Ns,(8i O s, (1 1/n)E) and thus (1 1/n)- Q ~ Xf (As) where
As = 2-1(1 1/n)-1(0, ,) E (0, 0,). Further, for any k > 0, we write


p~k(||D -(X i p,)||2 __jl) k[ 1 -)d < C2(l ~

+~~ ck p-_ -1 -1)Q > C2(1 + g)].






=~ E[Bi X B(X( X,)pc{(1l- n )BQ}]

x [8a- Xa B(Xi X,,)pc{(1 lljd)BQ]

=Ee ((Os X )(e, X,)'}

- BEe [{(Xi 04)(Xi X,)' + (Xi X,)(Xi 04)"}pc{(1i t n6- )BQ}]

+t B2Ey[(Xi X,)(X, X,) p {(1 + n- )BQ}].


(B-1)














(B-2)


We writ


Now,


Eg [(Xi 04)(Xi X,)Tpc{(1 + n- )BQ}]]

=Ee [{(Xi X) (0, 8,1)}(Xi X,)Tpc{(1i+ n2- )BQ}]

=Ee [{(Xi Xn) (0, 8,1)}(Xi X,)TI[(1 1/n)- Q < c2( Cjl

+c(1 + g) Eg [{(X, X,) (8i ,) }(Xi X,)7

x ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~( {1-1) } I(-1/)Q>c21g),(B3)

and the first of the two expectations in the last three lines of the above equation is

calculated by applying Lemrna A.1.2 with Y = Xi X,, rl = 8i Os, a = 1 1/n,









b = C2(1 + g), 4 = 0 and d = 0, while the second one is calculated by setting d = 1/2,

keeping the rest of the specifications same as before and reversing inequalities.

Similarly,


Ee[(Xi X ,)(Xi Xn) pPl {1n~- )BQ}]

=Eg [(Xi Xn)(Xi X,1) I[(1 1/nL)- Q < c2( gl

+tC2(1 + g)Ee [(Xi X,)(Xi X,)T

x{(1l 1/76)- Q}- I[(1 1/IL)- Q > c~2(1 +t g)l] (B-4)

and these two expectations are calculated using Lemma A.1.2 exactly as we did in

Equation B-3, with the only difference being that in the second of the two expectations
of the above equation we set d = 1 instead of d = 1/2. The result follows from combining

Equations B-2-B-4, and collecting the coefficients of E and (0, 0 s)(8i 8)T

separately. O

B.2 Proof of Theorem 3.5.1

We first write



= E{( Of"*+ Of* 0 E ) (0, Of" + Of"* 0 ) }

=P rz(*, Of"*) + llE{(B* 0 (")(Of"* 0CC Ti} (B-5)


Now,


Of*-i ( (XI XL)( {Bpc(||D- (X % Xn)||12) *}. (B-6)

Therefore

M~ E{(Of"B* 0 (")(Of"* 0 ) }


= [(X, X,)(Xi X,)T{Bpc(||D- (X i X,)||2) B* 2]. (B-7)









Note that Xi X

| |D35 Xi Xn) | |2

pc (| |D- T(X i X,) | 2
we write MI as


(1 1/n~~)5(B*) E Z where Z ~ N,(0, I,). Also that,

(B/B*) | |Z| |2. Using the last equality, it is easy to show that

c* (| |Z|12 | Where c* = c(B*/B) I. Returning to Equation B-7,


M~ = (1 1/n)(B*)- E E[ZZT {Bp.(||IZ)|| _" pa B2]E ,

and it follows from Equation 3-22 that


M~ = (1 1/n)B*E {(B/B*)pe. (U) 1}2 E,

where U ~ X@+2'

Combiningf Equations B-5 and B-9, we obtain the result.


(B-8)


(B-9)









APPENDIX C:
PROOF OF THEOREM 4 SERIES

C.1 Proof of Theorem 4.2.1

Proof. We first write

r1(( Of) =E{(0, O f")(0, O f") }

=E{(ei O f + Of Of") (0, Of + O Of") }

=(Is, B)E + E{(OfB Of ")( Of Of ) }. (C-1)

Next, with Of" = (Is, B)Xi + B Xn, we write



= E(Of Of" + O"- Of ~)(Of OfB + Of"-_ Of") }. (C-2)

It is easy to show that


E{(O e Of")(Of eE ) } = n- BE, (C-3)

and also that


E{(O f")(f" Of" } BE(p, X,(X, ,) B )}= 0 (C 4)

with the last equality following from the independence of X, and (Xi X,, S).

Finding anl expression? for E{(Of" -i O")(O~fi Of f)T} completes th~e ta~sk. First,

Of" Of" =BX )-B( 1)=(-B)X ), (C 5)

and thus


C; -- E {(OfB -Of)(OfC -O ) }

=E{(B B)(Xi X,)(Xi ,) (B B) }. (C-6)









Notice that


CI = C2 = .. = no -1C C
j=1


c -1E{(B B)S(B B)T}.


(C-7)


(n 1)(A + E) and E(S- )


Also, E(S)


(n p 2)-1(A + E)-l. Thus, by expansion,


E{(O" -OEB)(Of"


Of")T} = n- {n -- 1 -- 2a + a2(n p


2)- }BE.


(C-8)


The proof is completed by combining Equations C-1-C-4 and C-8.

C.2 Proof of Theorem 4.2.3

Proof. First, the frequentist risk of the OfE" is expressed as


R1(0s, Of")


Eg {(0, O f")(0, O f")'}


Eg[{0 X + B(X, X,)}{0 X + B(X


Xn)}"]


= + a2CE~y S-1(X,

+aEy ((Os Xi)(X,


x )(Xi x,)"S- }e


X,) S- }E + aE~y {S- (X4


Xn)(04 Xi)T}. (C-9)


We write X(-i)
we can see that


(X',..,XT XT ,.,XT)T and using the result of Lemma 4.2.2,


~Eg((Os X,)(Xi ,) S- }


Xi)(Xi 1,) S- |X -) }]


E [Ee {((8


-EE(8{S-l(X,


x,)}/X,) ,


(C-10)


which when combined with Equation C-9 gives us


R1(0s, Of") = E +

a~EEy{S-'(X4


Sa2CEE( S-1(X


x,)(Xi x,)"S- }e


1,)}/8Xs E. (C-11)


X,,)}/8X, + 8{(S-(Xs


Next, observe that


8{S- (Xs


{BS- /8Xi}(Xs


X,) + S- {8(Xs


X,)/8Xi}. (C-12)









Now, it is easy to see that


8(X, X,)/8X, = (1 1/n)I,.


(C-13)


Also, it is true that


8S-l/8X,


-S- {8S/8Xij}S- ,


(C-14)


-T
nX, X,,


where Xij is the jth element of vector Xi. Since S = CE= XkX


aS 8Xi 8X
X + X
8X,, 8X,j d X


XT
X '
X,,


8Xi -r
X
8X, "


8X'
(Xi X,) *
8X,


8Xi
+ (X4
8X,


X,)T.


(C-15)


From Equations C-14 and C-15,


8S-l /8Xi


-S- (Xi x) f


f (Xi Xn)7S-


(C-16)


where fi is the jth' c~olumnn of mnatrix S-


(fl ,..., fp). Now, from Equation C-16, we


see that


(aS- /8Xey)(X4


X,) =-S 1(Xi X,)(X, X,)7f,-


tr{S- (X, X,)(X4


Xn))}f ,


(C-17)


and thus


-trS- lx,(Xi x,)(>


-S- (Xi X,)(X, X,)TS-]


Xn))}S ]


(C-18)


The result of Equation C-18, along with Equations C-12 and C-13, shows that


8{S- (X4


S- (Xi X,)(X, X,)TS-1


tr {S- (Xi X,) (X, X,)"} S- + (1_ 1n) S-


(C-19)









The result of Theorem 4.2.3 follows from combining Equations C-11 and C-19. O

C.3 Proof of Theorem 4.4.1

Proof. Without loss of generality, we derive the B n-;-s risk of the nth limited translation

estimator, O ,E. We begin with the Helmert orthogonal transformation


Y1 = n 2(X1 + --+ X,) = zX,

Y2 = 2-E(X1 X2)



Y, = {n(n -1)}-T(X1 + --+ X,_ (n -1)X,)

-(1- /n)-(X,-X,).(C-20)


Then ,


i= 2
X, X, = -(1 1/n) Y,, (C-21)


and Ye N,(0, A + E), i = 2, .. ., n. Accordingly,


O, ~ = X, + (1 -1)BYp(|aSY|2,(C-22)

Of" = X, +(1 -1/n)7BY,. (C-23)

Next we calculated r1(( O )B,


r z((, 0 )" = E {(On 0 ) (On 0 ) }

= {(On -E Of +Of ijE) (0, -E 0 E +0 ) }. (C-24)


The B n-;-s risk of BLE is now written as

rz((, O ") = r((, Of") + (E{(B 0 B)(OfB 0 )


+E{( O")(O"- }E +( AE{(O -L~ 0 (0, -~ O)} (C-25)









We, continueI by conideing~ll E{(6 c, /V n )(6 -0 )}. Note that


.I = B(X,, p(|kS X ,)||12) _) _C267)

where~~~~ ki=(-1/)2 Also, writing as equal in distribution

S-r (X, (1) =n -( 1ni( YY -Y

i= 2
n-1
= (1 /~n) (~ Y;Y + Y,Y )-Y
i= 2

where W1 ~ W(I,, a 2, ) independently of Z ~ NV(0, I,). Further,


||S-i(Xll ,)||2 (X, XI)TS-'(X71 ,)

<2 (1 1n)Z (W 1+ ZZT)- Z =(1 -1/71)||(W + ZZ")-iZ||2 _28)

From Equations C-26-C-28, it follows that




=a2(1 1/n)E~(A + E):C E(WI ZZ")- ZZ (W1I+ ZZ )1

x {pc(a||(W1 + ZZ ) Z|_ 2 (A+E-E.(-9

We now continue with the calculation of


MEE[(WI +ZZ )- ZZ (WI ZZ )--

x {pc(a||I(W1 + ZZ)- Z||2 j2 =1,0 2M2,0 3,0 (C-30)

where, for k = 1, 2, 3 and I = 0, 1, Mk~~l are defined as


Mk~l- EE (WI+ ZZ')z-1ZZ (W + ZZ ) pk-1(a||(W +ZZ )-iZ||2))

For the time being, we need only Mk,0, k=1, 2, 3, but we will need M. i ,, k=1, 2, in

order to calculate the cross product, E{(0, 8C )( 8 BC)". ow lt









(W1 +ZZT)-iZ. Then the matrices UUT and W + ZZT are independently distributed

(Srivastava & K~hatri (1979), p. 95). This independence result allows us to simplify

matters. We first rewrite Mk~~l aS


Mk~ = E (W1+ ZZ")z- ~UU (W1+ ZZT)Ap -(a||U||2) _(:32)

Notice that ||U||2 __ tr T). Thus, alternatively, Mk~~l is Written aS


Mk~~l = E{(W1 + ZZ )I-, Hk(W1 + ZZT) 2}, (C-33)

where Hk, = E{UU p ~-l(a||U||2) NOW, the density of the random vector U is given by


fi;(u) =c*|I, -~1 us | I[U a < 1] =c*(1 -au) 2 I=P~IuTu < 1], (C-34)


(Srivastava & K~hatri (1979), p. 95), where I[.] is the indicator function and c* is the
normalizing constant. It follows that for i / j,


E{Ui Ujpp (a||IU||2")} = E{Ui(- Uj~p -(a|| IU||2"). _35)

The aboveP equ~ality holdtrue inp f andl only if E{Uippi n-] (a||U||2)} = 0, i / j. Further, we

have that

E{r72 ~-1 ll 12)}= --= E{U2-l~U1 2>

=~72 {(Ufp-] (a||U||2>
i= 1

= p (utu-p(autu) f(u~du do

-Jo1- oI, I, (utu)pI-l (aLUtu)(1-- usu) da. u C-3
fo o ;(1- u"U) np2 dU ..dU,

In order to evaluate the above integral we consider the polar transformation


ul = r sin Or, u2 = T COS 81 Sin 82, U3 = T COS 81 COS 82 Sin 83,

..., u,_l = r cos 01 ... cos Op-2 Sill 8p-1, up = T COS 81 .. COS 8p-2 COS 8p-1. (C 7)









The Jacobian of the transformation is r"- COSp-2 81 COSp-3 82 .co COS_- and anu = T2 I

follows that for any i = 1, .. ,p

pl~~~iLf k- k-/RJ1" 2 -p+1( 1.)2
U~~fo (r)P (11) r

1/o19k-1 1Beta(- ,"
pl pE p
Jio (r" (1 r) dr Beta(" "--) rc



where W ~ Beta((p 2)/2, (n p 1)/2). Thus, Hk, is a diagonal matrix with entries

(n 1)-1E {pi- (aW)} in its main diagonal, that is H, = (n l-1E {pi- (aW)} I,.

This, along with Equation C-33, implies that, for k = 1, 2, 3 and I = 0, 1,


Mk.l r 1E- { p% (a W) } E(W1 + ZZj")-'. (C-39)

Since (W1 + ZZT) ~ W (I a 1),


M~k,0l_ -( )-E{ a) o (C- 40)




Thus, returning to Equation C-30, we see that


M=(n -1)>-(n p- 2)- [1 2E(pc(aW)} +E~p 2(aW)}]I,

=(n 1)- (n p 2)- E{1 pc(aW)}2I, _42)

which when combined with Equation C-29 gives


E{(f"- )(O ) }=a2 -1(n p-2)-1E{1 p(aW)}2BC C3

Finally, we need to calculate


E{(08, Of)(O "nE ) } E {E(0, Of )(O "nE 0 ()T|X}










=E B (X,, X,) B(X,, p+ X, -X,,)

x (X, X,) B (c(||kIS-(X, Xn)||2)_)1(

Using the results of the orthogonal transformation, shown in the Equations C-21, we write

the above expectation as


E{O -0)(6 -~ e0 j)}

S(-1nEE{(Y Y )i/- -(A + )- }Y,Y n(~ Y Y )\]i
i= 2 i= 2

x {pc(|| ( ~YY -Yn ) ) Y||2
i= 2

+ (1 1/) E nY pY(~ Y Y)
i= 2

x {pc(||a ( Y Y )~~ Y|2) _) _45
i= 2

and due to the independence of the Yi, i = 1,..., n, and na 2E(Y1) = pw, the second of

the two terms of the last expression is equal to a matrix of zeros. We now continue with

the calculation of the first expectation in Equation C-45. Writing Zi = (A + E)-EYe,




i= 2 i= 2

x {pc(a|| ( fZZ )21 Z,||)-1} (A+E)-1
i= 2



Recalling the results of Equations C-40 and C-41 and combining Equations C-45 and

C-46 yields


E{(0, Of")(O "B ) } =B an- {1 a(n p 2)- } {1 Epc(aW)}BE, (C-47)

which for the choice of a = n p 2 is equal to a matrix of zeros.









The result of the Theorem follows from combining the result of Theorem 4.2.1 with

Equations C-25, C-43 and C-47. O

C.4 Proof of Theorem 4.5.1

Proof. Starting from Equation 4-20 and using the multivariate version of Stein's identity,
we write


Eg {(Bi Xi)(Xi ,) S- pc} = -CEE [8{S- (Xi X,)pc}/8Xi] (C-48)

Now, using the differentiation product rule, we obtain that

8[S- (Xi X,)pc] 8[S- (Xi ,)] 8ipc
pe S- (Xi X,) (C-49)
8iXi 8iXi 8XT'

We have provided an expression for 8[S-1(Xi X,)]/8Xi in Equation C-19 and we now

obtain an expression for 8pe/8Xi. Since the function pc is given as

P IllkS~(X X.,)I c] cl[||klS-a(Xi X,)|| > c]
||IklS- z(Xi X,)||

it follows that


8pc/8Xi = pal[||klS-T(Xi X,)|| > c], (C-51)

where


pd = c8{k (Xi X,)TS- (Xi x,)}-4/8X,
c: 8lax
||-TXX,)|| -3 {(Xi ,) S- (Xi X)}. (C-52)
2kl 8iXi

Using the differentiation product rule and the result of Equation C-16, we can show that


8Xy(Xi X,)S-1(Xi ,)T =/ 2 1 /M) fTy (X 1)
-2+T f -a:(Xi X,) (Xi ,) S- (Xi X,), (C-53)









where f,- is the jth column1 of S-l. Therefore

(Xi x,)S- (X, x,)' = 2(1 1/n)S- (X,
8Xi


x,)


2S- (Xi X,)(X,


x,)"S- (X, 1 ,.


(C-54)


Thus, by combining Equations C-51, C-52 and C-54, we obtain


1- n-l-||S-T(X,
X,)|| > c]
||S-a(Xi X,


Xn)||2


8~X


c
I[||Ik, S- (X,
k1


x (Xi ,) S-


(C-55)


Further, we combine the Equations C-55, C-49 and C-19 to obtain

8[S- (X, Xn)pc]/8Xi


-tr{S- (X, X,)(X,


X,)T}S- p + (1


1/n)S- pc


S- (Xi X,) (X, ,) S lpc


I[||kI1S 2(Xi-X,,)||>c]
Xn)||2


x (1 (


1/n)|| IS-a(X,


(C56)


Combiningf Equations C-56 with 4-20 and C-48 completes the proof of Theorem 4.5.1. O









APPENDIX D
PROOF OF THEOREM 5 SERIES
D.1 Proof of Theorem 5.2.1


Proof. First we write


rl((, Of"E)


E{0 fjE) (0, O") }


E{(eiB Of + O -Of) (0,~ Of + 0- OT)


(I,-B)E + {(Of


OfE")(O -~ Of B)T }


(D-1)


Now,

Of Of" = (,X)-BX )=(B B)(X, X.) B(X. C1), (D-2)


and it follows that


E{(Of Of)(Ofi Of) }


x.)(x ( X .) (B


B) }


E {(B B) ( X,


+ E {B(X. pw)(X. W) B }.

Marginally, Xi 1 N,(pw, E + A), i = 1,. ., n, and thus

E{B(X. pw)(X. pW) B } = n- BE.


(D-3)


(D-4)


X.) ( Xi X.)T(B B)T}. Notice that


We now calculate W


E {(B B) ( X;


W; = W -- n-l C1 W;


=n-1E{(B B)S(B B)T} Now,

E(BSBT) = a2b2k-2E(VS- V),


(D-5)


and


E(VS- V) = E{E(VS- V|V)}

(n p-2)- E{V(A + )- V}.


E{VE(S- )V}


(D-6)









Recall that V ~ W~l,(kE, df = n(k 1)), th~at is V d Ye~'l Ye "i where Y 'y

Nv,(0, kE). Also, note that Yi d (kE~) Zi where Zi N~1,(0, I,). We thus write
n(k-1) n(k-1)
E{V(A+ E)- V} = k2E ZZ)E(+E-E(yf
i= 1 j= 1

=k2E E CZiZE (A+E ,)- E ZyZ E

n(k-1)
+k2E Er( ZiZ ET C(A + )-1 EZ ZiZ E (D-7)
i= 1

and it is now easy to see that


E{V(A + E-) V = k2n(k -1){n(k -1) 1}BE

+k2n(k 1)CE EZZ E (A + E)- E ZZT}E (D-8)

where Z has the standard normal distribution. Let, DE E= (A + *-E. o h

expectation in the last line of D-8 is written as


Q E{ZZ DZZ } = E{(Z DZ)ZZ } = E() zizydijZZ ), (D-9)


where dij is the (i, j)th element of the matrix D, i, j = 1,. .p, and ze is the ith element Of

vector Z, i = 1, .. ,p. The kth diagonal element of matrix Q is calculated as


E~ ~ lk { zzdaI =E fdez)= rD 2dkk (D-10)


while the expression of the (k, 1)th, k / 1, element of matrix Q is obtained as


E{ (zzyda~zk~} = dk,(D-11)


since the matrix D is symmetric.

Combining Equations D-9-D-11, we obtain that


Q = 2D + tr(D)I,. (D-12)









Next, combiningf Equations D-8, D-9 and D-12 follows that


E(BSBT) = a2b2(n


p -2)-ln(k ) [{n(k ) + 1}B+ tr(B)z]E.


(D-13)


Now, from Equations D-5, D-6 and D-13 follows that


E(BSBT) = a2b2(n p 2)-ln(k


1) [{(k( -- 1) + 1}B + tr.(B)Iz,] .


(D-14)


Further, we have the following two results


E(BSBT)

E(BSBT)


(n 1)BE,


abn(k 1)BE.


(D-15)


From Equations D-14-D-15 follows that


E {(B B) (14


x.)(x ( X,.) (B B) }


=n- [a'b2(n; p- 2)- n(k 1) {n(k 1)$ 1+ 1}

+ a2b2(n p 2)- n(k 1)tr(B)E


1 2abn7(k 1)]BE


(D-16)


The result of the Theorem follows from combining Equations D-1, D-3, D-4 and D-16.


D.2 Proof of Theorem 5.3.1


Proof. W;lith~out loss of genecrality we calculate th~e Bwe~S risk of th~e estim~a~tor of 8,, 8if,~


r1((, 0, (")


E{(On, diC )(On 0, ) }


-
E {(On Of"E + Of"


ii E) (On i-E Of 0 n
-0n )(f 0


ef0, )


+E{(On Of"E)( ij

In order to calculate E{(jf"R Ojcf


- 0, )) }+ E{(Of E
c)(O 0 )}


ef, )(On Of") }. (D-17)


write


Oif"B ,("=(X, -1 ){c||i -( ,


x.)||2) 1},


(D-18)









11I
where pc(u) = min(1, c/ ), and kI = (1 1/n) za2.


Note that marginally Xi 1- Ns,(p, A + E), i


1,...,n. Consider now the Helmert


orthogonal transformation


Y1 = n 2(X1+ ---+ X,)


nMX.


1fx ,


{n(n -1)}- (X1 + ---+ X,_1


(n -1)X,)


1-
(1- 1/n)-T(X, X.).


(D-19)


Then


S = YsY y,y
i= 2


(D-20)


Also, Yi 1 N,(0,A+E), i = 2,..., n. Recallta f=akIS' hn


OEB E (1 -1/n?)iaXbk- VSI {,Pc(||IICS- Y~r 2) -1


(D-21)


and thus


E{(OiEB L iEB iEB
\V C,n \V


LEB T,'


(1 /lb)ar'b~lk-2EVS 1 Y YTS V {pc(| |aS; Y| |2


2 .


(D-22)


Since V is independent of (S, Y), the last expectation can be calculated as


E VE[S-' Y ,TS-' {pc(|| crS-A |2 "

In order to evaluate the inner expectation notice that


(D-23)


n-1
S- Ez = ( y,',d~ ) C( )-(WI ZZ )-'Z.
i= 2


(D-24)


2 ~]V .









Also, ||Sd Y,,|| ||(W1 + ZZ' -) Z||, where W1 ~ H (I,, a 2) independently of
Z ~Nz(0, Is,). Therefore


E [SIY, yS- {pc(||aInS-Y, 2 2)

=(A + E)- E(W, + ZZ )- ZZ (W, +ZZ')-'


x {pc(||lai(Wi + ZZT)-EZ||2


(D-25)


In Equation C-39 we showed that

Mk~l EE W1+ ZZT)'-'ZZ (W1+ ZZ )- pk-1(a||(W1+ ZZ")-$Z||2)


(n- )-E pk-(aW>) }E(W1 + ZZ")-,


(D-26)


where W ~ Beta((p+2)/2, (n-p-1)/2), k = 1, 2, 3 and I = 0, 1. Hence, combining Equations
D-22, D-23, D-25 and D-26, we obtain that


E{(OEBLEB IiEB


LEB T~u


a2 b2k- 22 -1(n p 2) 1E{ 1


pc(aW)}2E{V(A + E)- V}.


(D-27)


Using the result about E{V(A E )-1V}, given in Equation D-13, we obtain


x E{1 pc(aW) }2 {n(k


ii, )') }= a'b2(n p

-1) + 1}B + tr(B)] E.


2)-'(k 1)


(D-28)


The final step for calculating the Bayes risk is to provide an expression for the cross
product


To this end, we calculate


Of"}(Of"


diff ) ']. (D-29)


iifi ) } E[{E(0,|1 ,)


E(0,| 1,)


Oifi" = B(X. X.) -B(XII p X. + X.),


Oi e0, = B(X, X.) {pc(||k1S-(X,


(D-30)


2>' (A E)C-5


x.)||2) 1}.








Thus, in light of Equation D-20 and writing pc for pc(||kiSfr(Xs


X.)||2), Ln becomes


L, E-( /n Bx +(1- /n)B +B(p Y,))


1) .


-(1 1/n) (B) (pe


(D-31)


N~ow, a 2 -E( YI)


pw and Y1 is independent of ( Y,, S, V). Thus, L, is equal to


L, = (1 1/n)E { (B


B) YFy (B) (pe


1) } = L,,i + L7L,2,


(D-32)


where


L,,i_ U 2b~k-2(1 1/n1)E VS- YeyT S- V(pe

Ln,2 -abk- (1 1/n)BE( Y ,S-' V(p, 1

First,

L,,= a~b'k-2(1- 1/n)E VE{S' Y ,'S (pe -1)

Now, the inner expectation is written as


1) },


) }.


(D-33)


(D-34)


(A + )- E ([~ Z Z ) 'Z,,Z ( ZZ )-

x {pc(||a( Z Z j)- Z,||2) -1}] (A + Ej1


(D-35)


where Zi NV,(0, I,). Thus, combining Equations D-34 and D-35 with D-26 we obtain

L,,i a~b~k-2(1 1/n)E V(A C + E)-(2,0 111,0)(A + E)- V


a2b2k-2 -1(n


p -2)- E~pc(aW)


1}E{V(A + )- V}.


(D-36)


This along with Equation D-13 shows that

L,,i = a2b2(k -1)(n p- 2)- E~pc(aW)


1} [{n(k 1) + 1}B tr(B),]E. (D-37)


}V .









Similarly,


L,,29 = ab-(1- /)E(A + E)-iE E {Z,,Z ( Z Z )- {pe
i= 2

x (A + E)- V .

Using Equation D-26, we obtain the following expression


(D-38)


Ln,2 = -abk- (1 1/nL)E(A + E)-iE {(M~2,1


1,1)(A + E)- V }


ab(k 1)E~pc(aW)


1}BE.


(D-39)


Hence, combiningf Equations D-32, D-37 and D-39, we obtain


E((8, ~~H)(dif'L~


;jLEBdi~~",'>


ab(k 1)E~pc(aWI) 1} [ab(n


p 2)- {n(k 1) + 1} 1]BE


+a2b2(k 1)(n p 2) 1E~pc(aW) 1}tr(B)E.


(D-40)


n-l(k 1)-1. Then, combining Equations D-17, 5-9,


Let a, = n p 2 and be


D-28 and D-40, and collecting the terms of C, BC and tr(B)E separately, Theorem 5.3.1
follows. O


D.3 Proof of Theorem 5.4.1


Proof. Starting from Equation 5-17, we write


R1(0s, f"B)


= E + ab2k-2Ey {VS- (1, X.)(X,

+ bk;- Eg{(0, Xi)(Xi .) S- V}


x.)"S- V}


+ abk- Eg {VS-1(14


x.)(0, )"}'.


(D-41)


Consider now the following expectation


Ee{(0, X)(Xi .) S- V} = n6(k


xi)(xi .) S-'}E.(D-42)


1)kEy{((8









-(-i)
Write X


( .,X, 47,.. 1) and using the multivariate version of


Stein's identity, giben in Lemma 4.2.2, the expectation in the right hand site of Equation
D-42 can be written as


Xi)(Xi X.)TSII|X })]


Eg {(0, x)(xi X.)TS-'}


-~ey 8{S-'(14


(D-43)


The calculation of the derivatives can be achieved by using the product rule as follows


8{S- (As


{BS-'/ /814(14


x.) + S-'{(a14


X.)/8Xi}. (D-44)


It is easy to see that


8(Xi X.)/8 X


(1 1/n)I,.


(D-45)


Also, the following equality holds true


dS- /8Xsy


s-(Sl{8S/8sy}-',


(D-46)


-T -T
i Xm Xm 8 X. X. and


where Xij is the jth element of vector Xi. We write S = Cm


using the product rule again we see that


dS 8X r X
-X XT + Xe
ii~Xi- axji'


X.3X
83X ,


8Xi X
-iX j


-xT
(xi~8 -
(X, iX.) *


8Xi
+ x -(1 x.)".
8iXij


(D-47)


From Equations D-46 and D-47 follows that,


dS-l /8Xi


X.1'S-1


S- (1,x.)ff fj(As


(D-481


~E ["Eg((e


x.)}/814] .









where f,- is the jth column of matrix S-


(fl,... fp). Now, using Equation D-48, we


see that


(aS- /8Xey)(As


X.) = -S 1(Xi .)(Xi X.)7f3


tr{S-1(As


X.)(Xi X.) }f .


(D-49)


It follows that


(a-tr{S xi(X X


s-'(xi x.>(xi x.>'s-'


x.>


~.)(Xi x.)"}S


(D-50)


The result in Equation D-50, along with Equations D-44 and D-45, shows that


-tr {S- (As


x.>)lax,


s-'(xi x.>(xi x.>'s-'


x.) (xi x.)}S-' + (1 1/n)S-1


(D-511


Also, usingf similar reasoningf as in Equations D-6-D-12 we can show that


Eg {VS- (As


x.)(xi x.)"S-' V}


k~n.(k 1){n.(k 1) + 1}CEE( S- (As


x.)(xi x.)"S- }e


+ k"n(k 1tr Ey {S- (As


x.)(x, X.)TS-'} E.] C


(D-52)


The result of the Theorem follows from combining Equations D-41-D-43 and D-51-D-52.
O


D.4 Proof of Theorem 5.5.1


Proof. Starting from Equation 5-21, we write


Rl(0s, f")


E + a2b~k-2Ee {VS 1(As


x.)(xi x.) S 'Vp }~


+acbk- Ee { (Bi X4) (Xi .) S 1Vp-c)

+acbk- Ee { VS- (X, X.)(8i 1) pc}.


(D-53)









Using Stein's identity, provided in Lemma 4.2.2, we obtain an expression for


Ee{(0, Xi) (Xi X.) S

n(k )k~E( (0, X,)(14

-n7(k -- 1)kCEE [8{S-1(X,


~1Vp,)

- X.)7S- pe)C

X.)pc}/aXi] E.


(D-54)


We now continue by calculating the matrix derivative that appears in the last line of

Equation D-54. First, the differentiation product rule shows that

8 [S- (1i X.)pc] 8 [S- (1, X.)] 8ipc
pe + S-1(X, X.) (
8Xi 8Xi af"


D-55)


In order to calculate 8pc/8Xi, we write the function pc as

cl[||klS-a(Xs
pe = I[||klS-E(Xi X.)|| < c] +
||klS T(Xs


(D-56)


It follows that


1
dpc/8 Xi = pal [||IklS- z( Xi X.)|| > c],


(D-57)


where


p, = c8{k (Xi X.) S-1 (Xi X.))- }-/8X

2ki 8iXi


(D-58)






(D-59)


Further, it can be shown that


8Xij
-2fT(14


x.)'-'x S ( x, .) = ( /) .

- .)( i x.)"S- (xi x.),


and now it is easy to see that


8
(xi x.)"S- (x, x.) = 2(1
8iXi
-2S- (Xi X.)(X, X.)TS-1(14


1/n)S- (xi x.)

x.).


(D-60)


X.)|| > c]
-X.)||



































(D-63)


The result of the Theorem follows from Equations D-53, D-54, D-62 and D-63.


+ k:2n(k )r E S-(.


From Equations D-57, D-58 and D-60 follows that


1 n-l ||S-E(Xi X.)||2
X.)|| > c]
||S-(Xi X.)||3


8~X


c: _1
I[||IklS- z(X4
ki


x (Xi X.)TS-].


(D-61)


Combiningf Equations D-55, D-61 and D-51 we obtain that


S-tr{S- (xi .)(14

s-S-(Xi X.)(Xi x

lljS3x I


X.)T}S- pc + (1 1n


I [||kI S 2 (X X.)||I>c]


)S- pc


2 11"


.)


I


x


(D-62)


Further, similar calculations as in Equations D-6-D-12 show that


Eg{VS- (As


x.)(xi x.) S- Vp })


k2nL(k -- 1) {n(k -- 1) + 1}CEE( S- (As


x.)(xi .) S 'p2


X.)(X, X.) S- p }EE.










REFERENCES

BARANCHIK<, A. J. (1964). M~ultip~le Regre~ssion and Estimation of the Aferen of a M~ulti-
variate Nortmal Distribution. Stanford Univ. Technical Report No. 51.

Box, G. E. P. & Cox, D. R. (1964). An analysis of transformations. J. R. Strati~st. Soc.
B 26, 211-4:3.

CRESSIE, N. & READ, T. (1984). Multinomial goodness-of-fit tests. J. R. Strati~st. Soc. B
46, 440-64.

EFRON, B. & MORRIS, C. (1971). Limiting the risk of Bui-<< and empirical B~i-c
estimators-Part I: The B u-< < case. J. Am. Strati~st. A~ssoc. 66, 807-15.

EFRON, B. & MORRIS, C. (1972a). Limiting the risk of Bui-<< and empirical Bayes
estimators-Part II: The empirical B u-< < case. J. Am. Strati~st. A~ssoc. 67, 1:30-9.

EFRON, B. & MORRIS, C. (1972b). Empirical B~i-c < on vector observations: an extension
of Stein's method. Biometrika 59, :335-47.

EFRON, B. & MORRIS, C. (197:3). Stein's estimation rule and its competitors-an
empirical B is-< a approach. J. Am. Strati~st. A~ssoc. 68, 117-:30.

EFRON, B. & MORRIS, C. (1975). Data analysis using Stein's estimator and its
generalizations. J. Am. Strati~st. A~ssoc. 70, :311-19.

EFRON, B. & MORRIS, C. (1976). Multivariate empirical Bayes and estimation of
covariance matrices. Ann. Shetist. 4, 22-32.

HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J. & STAHEL, W. A. (1986).
Robust Stratistic~s. The Approach Based on Infll;,;. 0.. Functions. New York: Wiley.

HUBER, P. (1974). Robust Stratistic~s. New York: Wiley.

JAMES, W. & STEIN, C. (1961). Estimation with quadratic loss. Proc. 4th B~i. l~,7.1
Symp?. 1, :362-79.

SRIVASTAVA, M. S. & K(HATRI, C. G. (1979). An Introduction to M~ultivariate Stratistic~s.
New York: Elsevier, North-Holland.

STEIN, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann.
Strati~st. 9, 11:35-51.

ZELLNER, A. (1986). On assessing prior distributions and B li- Im regression analysis
with g-prior distributions. In Bar;;. -7.:;; Inference and Decision Techniques: E--r,t;- in
Honor of Bruno de Finetti, Ed. P. K(. Goel and A. Zellner, pp. 2:33-4:3. Amsterdam:
North-Holland.









BIOGRAPHICAL SKETCH

Georgios Papageorgion was born in Larnaca, Cyprus on January 29 of 1978. He

earned a bachelor's degree in Statistics from the Athens University of Economics and

Business in Athens, Greece, in 2000. After returning to Cyprus and working for one year

for a market research company, he decided to pursue graduate studies. He received a

Master of Statistics degree from the department of Statistics at the University of Florida

and a Ph.D. in Statistics in August of 2007.





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First,andforemost,Iwouldliketothankmyadvisor,Dr.MalayGhosh,forsharingwithmehisknowledge,fortheinvaluableguidanceheoeredmeandforhisencouragement.Iwouldalsoliketothankmycommitteemembers,Dr.RamonLittell,Dr.RonaldRandles,Dr.AndrewRosalskyandDr.MyronChang,fortheirconstructivecommentsandsupport.SpecialthanksalsogotothefacultymembersoftheDepartmentofStatisticsattheUniversityofFloridaaswellastoDr.JanetForresteroftheTuftsUniversityforprovidingmewithdatasetsandforhercollaborativework.Specialthanksarealsoduetomyfriendsfortheirsupportandencouragement.Lastly,andmostimportantly,Iwouldliketothankmyentirefamilyfortheirloveandsupport. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTIONANDLITERATUREREVIEW ................ 11 1.1Introduction ................................... 11 1.2OverviewofLimitedTranslationEstimators ................. 12 1.2.1TheBayesCase ............................. 12 1.2.2TheEmpiricalBayesCase ....................... 15 2MULTIVARIATELIMITEDTRANSLATIONBAYESESTIMATORS ..... 23 2.1BayesEstimators ................................ 23 2.2LimitedTranslationBayesEstimators ..................... 25 2.3BayesRiskoftheLimitedTranslationBayesEstimators .......... 26 2.4FrequentistRiskoftheLimitedTranslationBayesEstimators ....... 30 2.5RobustnessofLimitedTranslationBayesEstimators ............ 34 3MULTIVARIATELIMITEDTRANSLATIONEMPIRICALBAYESESTIMATORS:THECASEOFUNKNOWNPRIORMEAN ......................................... 41 3.1Bayes,EmpiricalBayesEstimatorsandInuenceFunctions ......... 41 3.2LimitedTranslationEmpiricalBayesEstimators ............... 44 3.3BayesRiskoftheLimitedTranslationEBEstimators ............ 45 3.4FrequentistRiskoftheLimitedTranslationEBEstimators ......... 48 3.5RobustnessofLimitedTranslationEBEstimators .............. 51 4MULTIVARIATEEMPIRICALBAYESANDLIMITEDTRANSLATIONEMPIRICALBAYESESTIMATORS ........................ 55 4.1Introduction ................................... 55 4.2EmpiricalBayesEstimators .......................... 57 4.3LimitedTranslationEmpiricalBayesEstimators ............... 60 4.4BayesRiskoftheLimitedTranslationEBEstimators ............ 61 4.5FrequentistRiskoftheLimitedTranslationEBEstimators ......... 64 4.6ASimulationStudy ............................... 65 5

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.................................... 75 5.1DevelopmentofEstimators ........................... 75 5.2BayesRiskoftheEBEstimators ....................... 77 5.3BayesRiskoftheLimitedTranslationEBEstimators ............ 78 5.4FrequentistRiskoftheEBEstimators .................... 80 5.5FrequentistRiskoftheLimitedTranslationEBEstimators ......... 81 5.6ASimulationStudy ............................... 82 5.7Application ................................... 85 6SUMMARYANDCONCLUSIONS ......................... 93 APPENDIX APROOFOFTHEOREM2SERIES ......................... 94 A.1TwoGeneralResults .............................. 94 A.2ProofofTheorem 2.4.1 ............................. 97 A.3ProofofCorollary 2.4.2 ............................. 100 A.4ProofofTheorem 2.5.1 ............................. 101 A.5ProofofCorollary 2.5.2 ............................. 103 BPROOFOFTHEOREM3SERIES ......................... 104 B.1ProofofTheorem 3.4.1 ............................. 104 B.2ProofofTheorem 3.5.1 ............................. 105 CPROOFOFTHEOREM4SERIES ......................... 107 C.1ProofofTheorem 4.2.1 ............................. 107 C.2ProofofTheorem 4.2.3 ............................. 108 C.3ProofofTheorem 4.4.1 ............................. 110 C.4ProofofTheorem 4.5.1 ............................. 115 DPROOFOFTHEOREM5SERIES ......................... 117 D.1ProofofTheorem 5.2.1 ............................. 117 D.2ProofofTheorem 5.3.1 ............................. 119 D.3ProofofTheorem 5.4.1 ............................. 123 D.4ProofofTheorem 5.5.1 ............................. 125 REFERENCES ....................................... 128 BIOGRAPHICALSKETCH ................................ 129 6

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Table page 2-1Minimumvalues,k,ofandP(k)forp=3 .................. 39 2-2Minimumvalues,k,ofandP(k)forp=5 .................. 39 2-3Minimumvalues,k,ofandP(k)forp=10 ................. 39 4-1Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=30. ........................ 70 4-2Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=30. ........................ 70 4-3Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=30. ............................ 70 4-4Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=30. ............................ 71 4-5Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=20. ........................ 73 4-6Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=20. ........................ 73 4-7Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=20. ............................ 73 4-8Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=20. ............................ 74 5-1Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=30. ........................ 88 5-2Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=30. ........................ 88 5-3Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=30. ............................ 88 5-4Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=30. ............................ 89 5-5Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=20. ........................ 89 5-6Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=20. ........................ 89 7

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............................ 90 5-8Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=20. ............................ 90 5-9MLE,EB,LTEstimates ............................... 91 8

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Figure page 2-1Plotof1scasafunctionofc. ........................... 38 2-2Plotofriskasfunctionofthenon-centralityparameter,. ............ 38 2-3Bayesriskundermisspeciedpriors. ........................ 40 3-1Risksplottedagainstthethenon-centralityparameter,i. ............ 54 3-2Bayesrisksplottedagainsttheassumedparametergwhenthetrueparameteristakentobeg=2. ................................. 54 4-1Plotof1scasafunctionofc,forp=2;10andn=10;30. ........... 68 4-2Theigeneratedfrom(a)thecontaminatedmodeland(b)thenormalmodel. 69 4-3Asampleofigeneratedfrom(a)thecontaminatedmodeland(b)thenormalmodel. ......................................... 72 5-1PlotofRLS=Ef1c(aW)g2asfunctionofc,forp=2;5andn=10;30. ... 87 5-2AverageintakesofvitaminsAandB1. ....................... 91 5-3EstimatedintakesofvitaminsAandB1. ...................... 92 9

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10

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Efron&Morris ( 1971 1972a 1975 ),andreferredtoas`limitedtranslationrules'bytheseauthors,isthesubjectmatterofthisdissertation.ThelimitedtranslationrulesarecompromisesbetweentheBayesandthemaximumlikelihoodestimatorsthatslightlyincreasetheBayesriskbutguardagainst 11

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Efron&Morris ( 1971 1972a 1975 )developedlimitedtranslationrulesfortheunivariatenormalcase.Theobjectivehereistodeveloplimitedtranslationrulesformultivariatenormalcase.Inthischapterwereviewtheliteraturerelatedtolimitedtranslationestimators. 1.2.1TheBayesCaseLetXjN(;1)andN(0;A).TheinteresthereisinestimatingtheunobservablequantitybasedonanobservationxonthevariableX.Underthesquarederrorlossfunction,L(;a)=(a)2,oranyotherincreasingfunctionofjaj,theBayesestimatorofgivenXis ^B(X)=A A+1X=(1B)X; whereB=(A+1)1.TheestimatingruleinEquation 1{1 isoptimalinthesensethatitminimizestheexpectedrisk,r(;^)=Ef^(X)g2,amongallchoicesofestimatingrules^(X),with 12

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1{1 is Clearly,ifthetruepriormean,isfarfromtheassumedpriormean,=0,thentheBayesriskof^Bisquitehigh.Also,thefrequentistrisk,thatistheriskasafunctionof,denotedbyR(;^B)andcalculatedas willbehighwhenisfarfromtheassumedpriormean,=0,thereasonbeingthat^Bshrinksthemaximumlikelihoodestimatorof,^0=X,towardsthepriormean.Ontheotherhand,^0=XhasminimaxriskequaltoR(;^0)=1forall,andthustheaverageriskof^0,theaveragetakenwithrespecttoprior,isr(;^0)=1which,however,isbiggerthantheaverageriskoftheBayesestimator,r(;^B)=1B.InordertocombinethegoodpropertiesoftheBayesrulewiththoseoftheMLE,thatistomaintainlowBayesriskandatthesametimeputaboundtothefrequentistrisk, Efron&Morris ( 1971 1972a 1975 )proposedlimitedtranslationrules.TheserulesaresimplecompromisesbetweentheBayesrulesandtheMLEs.Fortheestimationproblemathand,theyproposedaruleakintotheBayesrule,whichhoweverdoesnotallowdeviationsfrom^0=Xbiggerthanaxedvalue,saym. 13

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^LB=8>>>><>>>>:X+mifXd: Thelimitedtranslationrulecanalsobewrittenas ^LB=f1min(1;mB1 2 2)g^0+min(1;mB1 2 2)^B=f1c(BX2)g^0+c(BX2)^B=f1c(BX2)BgX; wherec=mB1 2andc(u)=min(1;c=p Efron&Morris ( 1971 )showedthat where1scisadecreasingconvexfunctionofc.Itiscalculatedas1sc=Ef1c(U)g2whereU23.Also,theseauthorsshowedthat supR(;^LB)=1+m2: TheBayesriskofthelimitedtranslationruleisaweightedaverageoftheBayesrisksoftheBayesruleandtheMLE,theweightsbeingscand1screspectively.This,ofcourse, 14

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1{3 anditisclearlyanunboundedfunctionof,whilethesupremumwithrespecttoofthelatteris1+m2. 1{1 ,thusresultinginanempiricalBayes(EB)estimator.TheEBestimatorshavebecomeverypopularsince Efron&Morris ( 1972a 1973 1975 )gavetheEBinterpretationofthecelebratedJames-Steinestimator, James&Stein ( 1961 ).Statisticianshaveappliedthesemethodstomanyimportantproblems,inparticularforsimultaneousestimationofseveralparameters.Considerthecasewherewehavep3independentunivariatenormalobservations where2=var(Xi)=1.LettingX=(X1;:::;Xp)Tand=(1;:::;p)T,thesamplingdistributionsin 1{8 canmorebrieywrittenasXjNp(;Ip),whereIpdenotestheidentitymatrixoforderp.Further,supposethat 15

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1{9 canbewrittenasNp(0;AIp).Thegoalhereistoestimatetheunknowni,i=1;2;:::;p.Therearetwocompetinglossfunctionsthatwetakeintoconsideration.First,fortheestimationofanindividualparameteri,weconsidertheindividualsquarederrorlossfunction whereaiistheestimateofthetrueparameteri.Wealsoconsidertheensemble,ortotal,lossfunctiongivenbyaddingtheindividuallosses wherea=(a1;:::;ap)Tisavectorguessfor=(1;:::;p)T.Underthedistributionalassumptionsstatedabove,theposteriormeanofigivenXi,thatistheBayesestimatorofi,is ^Bi=(1B)Xi: WemayrecallthatB=(1+A)1.SinceAisunknownsoisB,andtheBayesrulecannotbeusedassuch.However,Bcanbeestimatedfromthemarginaldistributionofthedata.MarginallyXNp(0;B1Ip),andthusBjjXjj22p.ItfollowsthatE(p2=jjXjj2)=B.Thus,^B=(p2)=jjXjj2isanunbiasedestimatorforB.SubstitutingthisexpressionforBinEquation 1{12 resultsinthecelebratedJames-Steinestimatorfori,namely ^EBi=(1p2 Invectornotations,^EB=(^EB1;:::;^EBp)T,iswrittenas^EB=(1^B)X. 16

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1{11 ,theriskoftheMLE,X,forall,is Underthesamelossfunction, James&Stein ( 1961 )showedtheriskoftheJames-Steinestimator,^EBis whereKPoisson(=21jjjj2.ThetermcurlybracketsintherighthandsideofEquation 1{15 isastrictlyincreasingconcavefunctionof.Ittakesonitsminimumvalueof2=pat=0anditapproachesitssupremumof1asincreases.Thus,^EBhasuniformlylowerriskthantheMLEforeveryvalueofinthesenseofthetotallossfunctionL(;a). Baranchik ( 1964 )consideredtheriskthatoccurswhenestimatinganindividualibytheJames-Steinestimator.Heshowedthattheriskof^EBiis whereKPoisson(=21jjjj2).TheriskinEquation 1{16 ismaximizedforxedjjjj2at2i=jjjj2,thatiswhenthevectorparameterhasallitscomponentsexcepttheithequaltozero,givingtheexpressionforthemaximumriskas 1+(p2)En2pKp+2 (p2+2K)(p+2K)o: Theexpressionin 1{17 attainsitsmaximumofapproximatelyp=4near=p=2.ThepointoftheaboveisthatalthoughtheJames-SteinestimatorhassmallertotalriskthantheMLE,aswesawinEquation 1{15 ,itmaydopoorlyinestimatingindividual 17

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1{11 averagedoverthejointdistributionofXand,isgivenbyr(;^EB)=p(1B)+2B.Therstterminthisexpression,p(1B),istheriskoftheBayesrule.Wecanthusthinkofthesecondtermasthethepriceforhavingtoestimatethepriorvariance,A,fromthedata.TheBayesriskoftheJames-Steinrulecanequivalentlybewrittenasr(;^EB)=pf1B(p2)=pg,wherethetermincurlybracketsisclearlylessthanoneandthustheBayesriskof^EBislessthanp,theBayesriskoftheMLE. Efron&Morris ( 1972a )developedlimitedtranslationempiricalBayesrules,acompromisebetweentheJames-SteinruleandtheMLE,inaneorttolowerthemaximumriskoftheindividualcomponentsoftheJames-SteinruleandatthesametimemaintainlowtotalriskandlowBayesrisk.ThelimitedtranslationrulefollowsascloselyaspossibletheJames-SteinruleprovidedthatitdoesnotdeviatefromtheMLEbymorethanaxedvalue.TheEB,orJames-Stein,estimatorwasobtainedastheestimatoroftheBayesrule,byreplacing(p2)=jjXjj2forB.Similarly,thelimitedtranslationEBruleisobtainedastheestimatorofthelimitedtranslationBayesrule.RecallthatinEquation 1{5 ,thelimitedtranslationBayesrulewasbrieywrittenas^LB=f1c(BX2)BgX,wherec(u)=min(1;c=p Invectornotations,^LEB=(^LEB1;:::;^LEBp)T.Sincetheargumentofthefunctioncisalwayslessthanorequaltop2,thevaluesofcthatneedtobeconsideredarethosebetween0andp 18

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TheweightoftheBayesriskoftheMLE,1sc,iscalculatedas1sc=E[1cf(p2)Wpg]2,whereWpBeta(3=2;(p1)=2),andforxedpitisadecreasingconvexfunctionofc.Thereforetheinversefunctionisalsodenedc(s)=c$sc(c)=sc.Forthetwoextremevaluesofc,0andp 19

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where andKPoisson().ThesphericalriskofthelimitedtranslationruleislessthanorequaltopsincethebracketedterminEquation 1{21 islessthanone.Theriskof^LEBi,asafunctionof,isnowexamined.Itisshownthattheriskofthelimitedtranslationestimatorofidependsonthrough2i=jjjj2andjjjj2.Wedenotethisriskby Forxedp,0c(s)p

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~R(p;sc)=supR(i;^LEBi)=fp;c(s)(1;p(s)2): Equivalently,~R(p;sc)isthesupremumoftheriskof^LEBioverallpriorson Forallvaluesofp,thebiggestreductionsin~R(p;sc),asafunctionofsc,occurnearsc=1.Thus~R(p;sc)canbeconsiderablyreducedbyincreasingtheBayesriskoftheBayesrulebyverylittle.Wenowdroptheassumptionsthatthevarianceofthesamplingdistribution,2,andthemeanofthepriordistribution,,areknown.WeconsiderindependentnormalmeasurementsXijjiindN(i;2),j=1;2;:::;k,whileiarethemselvesnormallydistributedvariablesiiidN(;2),i=1;2;:::;p.InthiscasetheBayesruleforestimatingiisgivenby ^Bi=XiXi whereXi=k1Pkj=1Xij,A=(k2)=2andB=(1+A)1.RecallthatthelimitedtranslationBayesrulefollowsascloselyaspossibletheBayesrulewithouthoweverallowingdeviationfromtheMLEbiggerthanaxedvaluem,say.Thatis,weimposetherestrictionthatjXi^BijmwhichisequivalenttojXijmB1.Thisrulecanbebrieywrittenas ^LBi=XiBcB(Xi)2(Xi)=+1BcB(Xi)2(Xi): Theunknownparameters;2and2needtobeestimatedfromthedata.First,fork>1,theconditionaldistributionsoftheXijgiveni,j=1;2;:::;kandi=1;2;:::;p, 21

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1 Thus,lettingS=Ppi=1Pkj=1(XijXi)2,itfollowsthat^2=S=[p(k1)]isunbiasedestimatorof2.Marginally,XiN(;2=k+2)=N(;2=(kB)).Hence whereX=(pk)1Ppi=1Pkj=1Xij.Itfollowsthatforp4 2=k=1 WenowwritetheunknownBthatappearsintheformulaofthelimitedtranslationBayesas 2=k: Anunbiasedestimatorofthersttermis^2=kwhileanunbiasedestimatorofthesecondtermis(p3)=V.AlsonotethattheunknownpriormeanisestimatedbyX.Replacingtheunknownparametersbytheirestimatorsresultsinthefollowingestimator ^LEBi=X+"1c(XiX)2(p3)^2 ThepropertiesofthisestimatorareexaminedanditisshownthatithasslightlybiggerBayesriskthattheregularEBestimatorbutitprotectsthestatisticianagainstlargefrequentistrisks. 22

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2.1 wereviewsomefactsconcerningthemultivariateBayesestimators.Insection 2.2 weintroducethelimitedtranslationestimators.InSection 2.3 weevaluatetheirBayesriskperformanceundertheassumedprior.InSection 2.4 weevaluatethefrequentistriskofboththeregularBayesandlimitedtranslationrules.InSection 2.5 wecomparetheBayesriskperformanceofthetwocompetingestimatorswhentheassumedpriordepartsfromthetrueprior.InSections 2.4 and 2.5 weconsideraspecicformofprior,theg-priororiginallyintroducedby Zellner ( 1986 ).SomeofthelongalgebraicderivationsareprovidedintheAppendixA. Thus,consideringthematrixlossfunction theBayesestimatoroftheunobservableisgivenbytheposteriormean ^B=X(A+)1(X)=(IpB)X+B; 23

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whichissmallerthantheBayesriskofthemaximumlikelihoodestimator(MLE),r1(;X)=.Muchinthespiritof Efron&Morris ( 1971 ),wenowcalculatetheBayesriskoftheBayesestimator^Bforanormaldistributionwithmeanvectorandvariance-covariancematrixA.LetNp(;A).Thenthetrueposteriormeanisgivenby^B=(IpB)X+B,whereB=(A+)1.TheBayesriskof^Bunderis whereEf(^B)(^B)Tg=(IpB).Wealsohavethat Thus,theBayesriskoftheBayesestimatorundermisspeciedpriorscanbeexpressedas Itisclearthatwhenthetruepriormean,,isfarfromtheassumedpriormean,,theBayesriskof^Bisquitehigh. 24

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Itisnoweasilyseenthatifisfarfrom,thefrequentistriskof^Bisquitehigh,thereasonbeingthat^BshrinkstheMLEof;^0=X,towardsthepriormean.Ontheotherhand^0=XhasminimaxriskequaltoR1(;^0)=forallandthustheBayesriskof^0isalsoequaltowhich,however,isbiggerthantheBayesriskoftheBayesestimator,r1(;^B)=(IpB),iftheassumedprioristhe`true'prior. ^LBc;i=Xi(A+)1 2hcf(A+)1 2(Xi)g; where isthemultidimensionalHuberfunction, Huber ( 1974 ),andcisaknownconstant. 25

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^LBc;i=XiB(Xi)c(jj(A+)1 2(Xi)jj2); where istermedtherelevancefunction, Efron&Morris ( 1971 1972a ).ThelimitedtranslationrulecanalsoberepresentedasaweightedaverageofthemaximumlikelihoodandtheBayesestimatorssince ^LBc;i=Xif1c(jj(A+)1 2(Xi)jj2)g+f(IpB)Xi+Bgc(jj(A+)1 2(Xi)jj2): MarginallytherandomvectorsXiNp(;A+).Thus,theargumentoftherelevancefunctionisthestandardizedsquarednormofXi,jj(A+)1 2(Xi)jj2.ThevalueoftherelevancefunctiondecreaseswiththeincreaseinthevalueofthestandardizedsquarednormofXi,thusreectingtheideathattherelevanceofthepopulationparameters,andA,isnotthesameforalli.WhentheobservedXihasahighstandardizedsquarednorm,theBayesestimator,andimplicitlythepriorparameters,isnotconsideredtobeveryrelevantforthecorrespondingi.InsuchacasetheMLEisconsideredtobemorerelevantandthustheshrinkagetowardsthepriormeanisappropriatelycontrolled.Inthesubsequentsectionsofthischapterwewilldropthesuxiandworkwithageneric^LBc. 26

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Thetermr1(;X)r1(;^B)isthesavings,inBayesrisksense,thatoccurwhenusingtheBayesestimatorinsteadoftheMLE,whiler1(;^)r1(;^B)isthelossthatoccurswhenusing^insteadoftheBayesestimator.Thegeneralizedrelativesavingslossof^LBcisgivenby andforthespecialcasewherec(u)=min(1;c=p 1sc=P(2p+2>c2)cp 2) (p+2 2)P(2p+1>c2)+c2 wherethecrossproducttermsdonotappearsinceE(jX)=^Bandthus 27

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2{1 that WenowneedtocalculateEf(^B^LBc)(^B^LBc)Tg.Notethat ^B^LBc=B(X)f1c(jj(A+)1 2(X)jj2)g; andwethushave 2(X)jj2)g2]BT: LetZ=(A+)1 2(X)Np(0;Ip),anditfollowsthat 2EfZZT[1c(jjZjj2)]2g(A+)1 2: ThefollowinglemmasimpliesthecalculationoftheBayesrisk. Proof. WenowcontinuewiththecalculationoftheBayesrisk.ByLemma 2.3.2 AgainbyLemma 2.3.2 ,wehave 28

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SincejjZjj22p, 21 2(p+2 2)dy=pEf[1c(U)]2g whereU2p+2.ItfollowsfromEquations 2{24 2{26 and 2{27 that Equations 2{18 2{20 2{23 and 2{28 showthat where1sc=Ef1c(U)g2. Now,bychoosingc(U)=min(1;c=p 2I(U>c2)g+c2EfU1I(U>c2)g=P(2p+2>c2)cp 2) (p+2 2)P(2p+1>c2)+p1c2P(2p>c2); which,forxedpdependsonlyoncanditisindependentofthemodelparameters.Further,ifwefeelthattheBayesruleisirrelevantforobservationsthathavestandardizednormbiggerthansomevaluec0say,c0>c,wecanmodifytherelevance 29

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Thecostofthismodicationintermsofincreasedgeneralizedrelativesavingslossis 2I(U>c20)gc2EfU1I(U>c20)g=cp 2) (p+2 2)P(2p+1>c20)p1c2P(2p>c20): WehaveseenthattheBayesriskofthelimitedtranslationruleisaweightedaverageoftheBayesrisksoftheBayesruleandtheMLE,theweightsbeingscand1screspectively,whichcausesageneralizedrelativesavingslossof(1sc)Ip.Also,theweightoftheBayesriskoftheMLE,1sc,forxedp,isadecreasingconvexfunctionofc.ThisallowsthestatisticiantochoosecbydecidingbywhatproportionitisworthincreasingtheBayesriskoftheBayesruleinordertoreceiveprotectionagainstlargefrequentistrisks. 30

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Zellner ( 1986 ),arecalledg-priors.Undertheassumedmodel,BreducestoB=(1+g)1IpandtheBayesestimatorisgivenby^B=X(1+g)1(X).ThefrequentistriskassociatedwithitisobtainedfromEquation 2{8 Also,thelimitedtranslationestimatorisisgivenby ^LBc=X(1+g)1(X)c(jj(1+g)1 21 2(X)jj2): WewouldliketocomparethefrequentistriskoftheBayesestimatortothefrequentistriskofthelimitedtranslationestimator.AnexpressionofthelatterisprovidedbythefollowingTheorem. 2Ef[2p+2()]1 2I[2p+2()>c2(1+g)]g2c(1+g)1 2Ef[2p+4()]1 2I[2p+4()>c2(1+g)]gi+h(1+2g)(1+g)2P[2p+2()>c2(1+g)]+c2(1+g)1Ef[2p+2()]1I[2p+2()>c2(1+g)]g2c(1+g)1 2Ef[2p+2()]1 2I[2p+2()>c2(1+g)]gi; 31

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A.2 ofAppendixA.SincetherisksinEquations 2{33 and 2{35 involvematrices,inordertographicallycomparethem,weconsiderscalarversionsofthem.Specically,weconsiderthequadraticlossfunction ItiseasytoshowusingEquation 2{33 thattheriskoftheBayesruleunderthelossfunctionL2isequalto ThefollowingCorollaryprovidesanexpressionfortheriskofthelimitedtranslationBayesestimatorunderthelossfunctionL2. 2h2Ef[2p+2()]1 2I[2p+2()>c2(1+g)]gEf[2p()]1 2I[2p()>c2(1+g)]gi: Theproofisgiveninsection A.3 ofAppendixA.ThebracketedterminthelasttwolinesofEquation 2{38 canbecalculatedas (p+2k p+2k1): Theriskof^LBc,forxedpandg,quiteconveniently,isafunctiononlyofthenon-centralityparameter=()T1()=2andsoistheriskinEquation 2{37 32

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2.5 weseehow1scdecreasesascincreasesforthreedierentvaluesofp=3;5and10andforxedg=2.Forp=3and1sc=10%thecorrespondingvalueofcis1:52.InFigure 2-2 weseehowtherisksinEquations 2{37 and 2{38 behaveasthenon-centralityparameterincreases.Forsmallvaluesof,i.e.whenisclosetothepopulationmean,theBayesrulehasslightlysmallerfrequentistriskthanthelimitedtranslationBayesrule.However,thefrequentistriskoftheBayesruleincreaseslinearlywiththenon-centralityparameterwhichclearlymeansthattheBayesrulehashighriskwhentheisfarfrom.Onthecontrary,thefrequentistriskofthelimitedtranslationBayesrulebecomesatafterexceedsacertainvalue.Thatis,thelimitedtranslationruledoesnotallowlargefrequentistriskseveniftheunobservableisfarfromthepriormean.ReturningtoEquation 2{38 ,wewriteR2(;^LBc)=R2(;^B)+ep;c;g().Theproposedestimator^LBc,doesbetterthantheBayesestimatorwhenthefunctionep;c;g()takesonnegativevalues.This,ingeneral,happenswhenattemptingtoestimatearandomeectwhichdepartswidelyfromtheassumedpriormean,thatis,whenthenon-centralityparametertakesonlargevalues.Thequestionsofinterestarewhatvaluesmusttake,forxedvaluesofp,candg,inorderforthefunctionep;c;g()tobecomenegative,andhowlikelythosevaluesare.WeattempttopartlyanswerthisquestionbyprovidinginTables 2-1 2-2 and 2-3 theminimumvalues,kofneededinorderforep;c;g()totakenegativevalues,forxedp,candg.Wealsoprovidetheprobabilitiesthattakesavalueasbigorbiggerthank. 33

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Table 2-1 showsthevalueskandthecorrespondingprobabilitiesP(k)forthecasewherethedimensionisp=3,forvedierentvaluesofthepriorparametergandforthreevaluesofc.Wemayrecallthatcand1scareonetoonefunctionsandthustheTableprovidesthegeneralizedrelativesavingsloss,1sc,alongwiththecorrespondingc.ObservingtherstrowofTable 2-1 ,itisclearthatforallvaluesofg,P(k)isbiggerthat1%,thegeneralizedrelativesavingsloss.Thatis,bysacricing1%oftheBayesrisk,wehavefairlybigreturnsintermsofthefrequentistrisk.SimilararetheresultsdisplayedonthesecondrowofTable 2-1 .Thegeneralizedrelativesavingslossis5%whilethereturnsinfrequentistriskarebiggerthan5%forallvaluesofg.Forthecasewhere1sc=10%,thereturnsinfrequentistriskarebiggerthan10%forg=2;5and10andsmallerthan10%forg=0:5and1.This,however,isnotdiscouragingbecausethereportedpercentages,P(k),arecalculatedassumingthattheprioristhetrueone.WecanexpecttheprobabilitiesP(k)toincreasewiththeincreasingdistanceoffromthetrueprior.TheresultsofTables 2-2 and 2-3 ,wherewehavechosenp=5andp=10respectively,aresimilar.Wehavefairlybigreturnsinfrequentistriskwhensacricing1sc=1%and5%oftheBayesrisk.Thereturnsinfrequentistriskwhensacricingfor1sc=10%oftheBayesriskarebiggerthat10%forg=2;5and10buttheyaresmallerthan10%forg=0:5and1. 2.4 .Thatis,XjNp(;)andNp(;g). 34

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2{2 ,itsBayesriskis UnderL2,thequadraticlossfunctionof 2{36 ,theBayesriskoftheBayesruleis NowsupposethatthetruepriorisN(;g).ThentheBayesriskof^BundertheL1andL2lossesis and respectively,where=21(1+g)1()T1().When=0andg6=g,thatiswhenthepriormeanhasbeencorrectlyspeciedbutghasbeenoverorunderestimated,theBayesriskr2(;^B),overorunderestimatesthetrueBayesrisk,r2(;^B).Also,r2(;^B)increaseslinearlywithandthusr2(;^B)underestimatesthetrueBayesriskwhenthepriormeanismisspecied.WenowconsiderthelimitedtranslationBayesrule.Undertheassumedmodelthelimitedtranslationestimatorisgivenby ^LBc=X(1+g)1(X)c(jj(1+g)1 21 2(X)jj2); 35

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2.3.1 itiseasytoseethat 1+g)sc=(1sc whichimpliesthat ThefollowingtheoremprovidesanexpressionfortheBayesriskunderpriorofthelimitedtranslationBayesruleobtainedundertheassumedprior. 2nEf[2p+4()]1 2I[2p+4()>c2d]gEf[2p+2()]1 2I[2p+2()>c2d]goi+(1+g)1h(g2g1)(1+g)1P[2p+2()>c2d]+cEnc[2p+2()]12d1 2[2p+2()]1 2I[2p+2()>c2d]oi; A.4 ofAppendixA,whileinsection A.5 weprovethefollowingresult. 2(1+g)1 2E[2p()]1 2I[2p()>c2d]+4c(1+g)1 2(1+g)1 2E[2p+2()]1 2I[2p+2()>c2d]: 36

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23 2c(1+g)1 2(1+g)1 21Xk=0ek (p+2k p+2k1g: Wenowrevisittheexampleoftheprevioussectionwherewesupposedthatwearegivennobservationsofdimensionp=3andbasedonourpriorbeliefswesetg=2.Thechoiceofc=1:52correspondstogeneralizedrelativesavingslossof1sc=10%.Whenthepriorparametersarecorrectlyspecied,theBayesriskoftheBayesruleandofthelimitedtranslationrulearer2(;^B)=2andr2(;^LBc)=2:1respectively.InFigure 2-3 (a)weplottheriskfunctionsr2(;^B)andr2(;^LBc)forvaluesofthenon-centralityparameterrangingfrom0to15andassumingthatg=g=2.Inthesamegraphweplotr2(;^B)andr2(;^LBc)which,however,donotdependon.Weseethatforverysmallvaluesof,theBayesrulehassmallerriskthanthelimitedtranslationrule.However,theBayesriskoftheBayesruleincreaseslinearlywithwhiletheBayesriskofthelimitedtranslationruleincreasesinamuchsmallerrate.InFigure 2-3 (b)weplotthesamefourriskfunctionsforvaluesofgrangingfrom0:2to10andfor=0,thatisassumingthatthepriormeaniscorrectlyspecied.Weseethatforvaluesofgclosetothetruevalue,g=2,r2(;^B)islessthanr2(;^LBc).However,whengisunderestimatedthelimitedtranslationruledoesbetterthattheregularBayesestimator.Astheassumedvalueg,ofgbecomesbiggerthanthetruevalueofg,theBayesriskperformanceofthetwoestimatorsbecomessimilar.Asgincreases,thetwoestimatorsbecomeclosertotheMLEandtheirBayesrisktendstotheBayesriskoftheMLE,r2(;Xi)=p,andherewehavetakenp=3. 37

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Plotof1scasafunctionofc. Figure2-2. Plotofriskasfunctionofthenon-centralityparameter,. 38

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Minimumvalues,k,ofandP(k)forp=3 g 0:5 1 2 5 10 1.79 3.81 8.27 22.14 44.941sc=1% 6.70% 5.45% 4.07% 3.13% 2.95% 1.72 3.32 6.72 16.96 33.601sc=5% 7.58% 8.43% 8.14% 7.91% 8.14% 1.76 3.21 6.23 15.21 29.841sc=10% 7.06% 9.29% 10.09% 10.76% 11.32% Table2-2. Minimumvalues,k,ofandP(k)forp=5 g 0:5 1 2 5 10 2.49 5.12 10.80 28.31 57.111sc=1% 7.64% 6.87% 5.55% 4.53% 4.36% 2.51 4.69 9.24 22.88 45.191sc=5% 7.41% 9.48% 9.99% 10.32% 10.76% 2.63 4.68 8.88 21.38 41.981sc=10% 6.17% 9.55% 11.39% 12.83% 13.57% Table2-3. Minimumvalues,k,ofandP(k)forp=10 g 0:5 1 2 5 10 4.17 8.24 16.82 42.98 86.051sc=1% 8.18% 8.70% 7.84% 7.02% 6.98% 4.41 8.03 15.42 37.57 74.211sc=5% 6.13% 9.79% 11.75% 13.10% 13.79% 4.77 8.33 15.53 37.10 73.011sc=10% 3.93% 8.22% 11.39% 13.80% 15.72% 39

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(b)Figure2-3. Bayesriskundermisspeciedpriors.`LBTruePrior'referstor2(;^LBc),`BayesTruePrior'referstor2(;^B),`LBAssumedPrior'referstor2(;^LBc),`BayesAssumedPrior'referstor2(;^B).(a)Bayesriskasfunctionofthenon-centralityparameterand(b)Bayesriskasfunctionofg. 40

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3.1 webrieyreviewtheBayesandempiricalBayesestimatorsaswellasthenotionofinuencefunctions.Insection 3.2 weintroducethelimitedtranslationestimatorsandinSection 3.3 weevaluatetheirBayesriskperformanceundertheassumedprior.Section 3.4 evaluatestheirfrequentistriskperformancewhileinSection 3.5 wecomparetheBayesriskperformanceofthetwocompetingestimatorsassumingmisspecicationofthepriordistribution.SomeofthelongalgebraicderivationsareprovidedintheAppendixB. Itfollowsthat 41

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whereB=(A+)1.Further,theposteriorvariance,alsotheBayesriskoftheBayesestimator,canbewrittenasvar(ijXi)=(A+)1=(IpB).SinceisunknowntheBayesestimator,^Bi=(IpB)Xi+B,cannotbeusedassuch.However,marginallytherandomvectorsXihavemean,andthustheunknownpriormeancanbereplacedbyanunbiasedestimator,Xn=n1Pni=Xi,thusresultinginanempiricalBayes(EB)estimator ^EBi=(IpB)Xi+BXn; ofi.ThesameestimatorcanbeobtainedasahierarchicalBayesestimatorwhenoneassignsauniformpriordistributionon.TheEBestimatorshrinkseverymaximumlikelihoodestimator(MLE),Xi,towardsthegrandmean,Xn,theMLEoftheunknownpriormean,.Indoingso,itattainsalowerBayesriskthantheMLE,undertheassumedprior.However,itresultsinhighBayesriskwhenthepriordistributionismisspecied.Italsoresultsinhighfrequentistriskwhenattemptingtoestimateparameters,i,thatarefarfromthegrandmean.Ontheotherhand,theMLEhasminimaxriskequaltoforalli.Inordertoavoidthesetwoproblems,wedeveloprobustEBestimators,namelythelimitedtranslationEBestimators.WestartbyassigningthenoninformativeprioronUniform(
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1984 ).Letf1andf2denotetwodensityfunctions.Thenthegeneraldivergencemeasureisgivenby Here,f1andf2denotetheposteriordensitiesofgivenX=(XT1;:::;XTn)TandX(i)=(XT1;:::;XTi1;XTi+1;:::;XTn)Trespectively.Inordertondf1andf2,notethatXijiidNp(;+A),i=1;:::;n,andUniform(
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3{7 wehave12=XnX(i)n1=(n1)1(XiXn)and(1+)21=(+A)(n+)n1(n1)1.Itisnoweasytoseethatthedivergencemeasureisaonetoonefunctionwith (XiXn)Tn(+A)1 whichisaquadraticformin(XiXn).BasedonthisresultwewillobtainsomerobustBayesianestimatorsinthefollowingsection. 2D1 2(XiXn). ^LEBc;i=XiBD1 2hcfD1 2(XiXn)g; where isthemultidimensionalHuberfunction, Hampel ( 1986 ),andcisaknownconstant.TheproposedestimatorcanequivalentlybewrittenasaweightedaverageoftheMLEandEBestimatorsince ^LEBc;i=XiB(XiXn)c(jjD1 2(XiXn)jj2)=Xif1c(jjD1 2(XiXn)jj2)g+^EBic(jjD1 2(XiXn)jj2); 44

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2(XiXn)jj,doesnotexceedacertainvalue,csay.Whenthisdistancetakesonavaluebiggerthanc,therelevancefunctiontakesonavaluesmallerthanone,andbythesecondlineof 3{13 weseethatthelimitedtranslationEBestimatorgivestheMLEpositiveweightattheexpenseoftheweightoftheEBestimatorandasthedistanceofXitoXnincreasesthelessrelevantisconsideredtobetheEBrulefortheestimationofthecorrespondingi.Inthenextsectionsweshowthatthisprovidesthestatisticianwithprotectionagainstlargevaluesofthefrequentistrisk,whileslightlyincreasingtheBayesrisk. Comparingr1(;^EBi)tor1(;^Bi),giveninEquation 2{4 ,weseethatthepriceforhavingtoestimatefromthedataisn1Bwhichconvergestoamatrixofzerosasnincreases,therateofconvergencebeingn1.Thisisintuitivelyclearsinceasthesamplesizeincreases,thesamplemeanXnconvergestothepopulationmarginalmean.Thefollowingtheoremgivesanexpressionforr1(;^LEBc;i).Thecalculationsdonotdependonthespecialnatureoftherelevancefunctionc(:). 45

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(3{16) iscalculatedasGRSL(^LEBc;i;Xi)=(1sc)Ip.Ifwechoosec(u)=min(1;c=p 1sc=P(2p+2>c2)cp 2) (p+2 2)P(2p+1>c2)+p1c2P(2p>c2); where(:)denotesthegammafunction. Notingthat ^EBi^LEBc;i=B(XiXn)fc(jjD1 2(XiXn)jj2)1g; itfollows,fromtheindependenceofXiXnandXn,andthefactthatE(Xn)=,that 2(XiXn)jj2)1g(XiXn)(Xn)T]BT=0: 46

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2(XiXn)jj2)1]2gBT=(11=n)(A+)1 2EfZZT[c(jjZjj2)1]2g(A+)1 2; whereZNp(0;Ip).ItwasshowninEquation 2{28 that whereU2p+2.Hence,fromEquations 3{18 3{20 3{21 and 3{22 followsthat where1sc=Ef[1c(U)]2g.ThesecondofthetwotermscanbethoughtofasthepriceintermsofincreasedBayesriskforlimitingthefrequentistriskoftheEBestimator.Alternativelywecanwrite thuscompletingtheproofoftheTheorem. TheBayesriskofthelimitedtranslationEBestimatorisaweightedaverageoftheBayesriskoftheEBruleandtheBayesriskoftheMLE,theweightsbeingscand1screspectively.Thiscausesalossinthegeneralizedsavingsof(1sc)Ip.However,theweightoftheBayesriskoftheMLE,1sc,forxedp,isadecreasingconvexfunctionofc.Thus,thechoiceofcisequivalenttodecidingbywhatproportionitisworthincreasingtheBayesriskoftheEBestimatorinordertoreceiveprotectionagainstlargefrequentistrisks.ThisprotectiondoesnotrequireincreasingtheBayesriskbymorethan10%. 47

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Now,XiXnjindN(in;(11=n)),wheren=n1Pni=1i.Itfollowsthat andthattheexpectationEf(iXi)(XiXn)Tgisequalto Thus,combiningEquations 3{25 3{27 ,weobtain whichundertheassumptionthatA=greducesto 11=n(1+2g) (1+g)2]: 48

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2Eif[2p+2(i)]1 2I[2p+2(i)>c2(1+g)]g2c(1+g)1 2Eif[2p+4(i)]1 2I[2p+4(i)>c2(1+g)]gi+(11=n)h(1+2g)(1+g)2P[2p+2(i)>c2(1+g)]+c2(1+g)1Eif[2p+2(i)]1I[2p+2(i)>c2(1+g)]g2c(1+g)1 2Eif[2p+2(i)]1 2I[2p+2(i)>c2(1+g)]gi B.1 ofAppendixB.Foreasiercomparisonoftherisksof 3{29 and 3{30 ,wecalculatetheirscalarversionsbyconsideringtheL2lossfunction First,itiseasytoshowthattheriskoftheEB,ruleunderthelossfunctionL2,isequalto which,forxedpandg,isafunctiononlyofthenon-centralityparameteri,andsoistheriskofthelimitedtranslationestimator,asbecomesevidentinthefollowingCorollary. 49

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(1+g)2g+2i(11=n)(1+2g)(1+g)2P[2p+4(i)>c2(1+g)]+c2(11=n)(1+g)1P[2p(i)>c2(1+g)]+2c(11=n)(1+g)1 2h2iEif[2p+2(i)]1 2I[2p+2(i)>c2(1+g)]gEif[2p(i)]1 2I[2p(i)>c2(1+g)]gi: TheproofisverysimilartothatofCorollary 2.4.2 ,giveninsection A.3 ofAppendixA,andthusomitted.ThebracketedterminthelasttwolinesofEquation 3{33 canbecalculatedas (p+2k p+2k1): Wenowconsiderthehypotheticalscenariowherethestatisticianisgivennobservationsofdimensionp=3.Wealsosupposethatnislargeenoughtoignorethe1=ntermsintheriskfunctionsin 3{32 and 3{33 .Alsosupposethatg=2andthatthestatisticianiswillingtohaveageneralizedrelativesavingslossof1sc=10%inordertoreceiveprotectionagainstlargefrequentistrisks.Forp=3and1sc=10%thecorrespondingvalueofcis1:52.InFigure 3-1 weseehowtheriskfunctionsof 3{33 and 3{32 behaveasthenon-centralityparameter,iincreases.Forsmallvaluesofi,i.e.wheniiscloseton,theEBestimatorhasslightlysmallerfrequentistriskthanthelimitedtranslationEBestimator.However,thefrequentistriskoftheEBestimatorincreaseslinearlywiththenon-centralityparameterwhichclearlymeansthattheEBestimatorhashighriskwhentheiisfarfrom.Onthecontrary,thefrequentistriskofthelimitedtranslationEBestimator 50

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where andthus 51

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^LEBc;i=XiB(XiXn)c(jjD1 2(XiXn)jj2); whereD=var(XiXn)=(11=n)B1.ItsBayesriskunderpriorisgiveninthefollowingTheorem. 2) (p+2 2)P[2p+1>(c)2]+p1(c)2P[2p>(c)2]: Theproofisgiveninsection B.2 ofAppendixB.WenowconsidertheL2lossfunction,givenin 3{31 ,andprovideexpressionsfortheBayesrisksofthetwoestimators.First,theBayesriskoftheEBestimatorisgivenby 52

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Figure 3-2 showshowthetwofunctionsbehaveastheassumedpriorparameterg,variesaroundthetruepriorparameterg,which,forthesakeofcomparison,wetaketobeg=2.Wealsotakep=3andntobelargeenoughtoapproximate1n11.Whenthetrueparametergisassumedtotakeanyvaluesmallerthan1:34,^LEBc;idoesmuchbetterthat^EBi.Thatis,whengisunderestimated,thelimitedtranslationestimatorhasmuchsmallerBayesriskthetheEBestimator.When,however,theassumedpriorparameterisclosetothetrueparameter,theEBestimator,asoneshouldexpect,faresbetterthanthelimitedtranslationestimator.Astheassumedvalueg,ofgbecomesbiggerthanthetruevalueofg,theBayesriskperformanceofthetwoestimatorsbecomessimilar.Asgincreases,thetwoestimatorsbecomeclosertotheMLEandtheirBayesrisktendstotheBayesriskoftheMLE,r2(;Xi)=p,andherewehavetakenp=3. 53

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Risksplottedagainstthethenon-centralityparameter,i. Figure3-2. Bayesrisksplottedagainsttheassumedparametergwhenthetrueparameteristakentobeg=2. 54

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55

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Efron&Morris ( 1972a )developedsomerobustestimatorswhichtheyreferredtoas`limitedtranslationrules'.TheserulesarecompromisesbetweentheEBandtheMLestimatorsthatslightlyincreasetheBayesriskbutguardagainstlargefrequentistrisks. Efron&Morris ( 1972a )developedlimitedtranslationEBrulesfortheunivariatenormalcase.TheobjectivehereistodeveloplimitedtranslationEBrulesformultivariatenormalcase.OneofthevirtuesofthelimitedtranslationrulesisthattheydonotfaretoobadlyintheirBayesriskperformance,comparedtotheregularEBestimators,eveniftheassumedpriorisclosetothetrueone.Inafrequentistrisksense,thelimitedtranslationestimatorsdonotperformtoobadlyrelativetotheregularEBestimatorseveniftherandomeecttobeestimatedisclosetothesyntheticmeantowardswhichtheMLestimatorsarepulled.Ontheotherhand,iftherandomeecttobeestimatedisfarfromthissyntheticmean,thenthelimitedtranslationestimatorsdoperformmuchbetterthantheregularEBestimators.Theorganizationoftheremainingsectionsisasfollows.InSection 4.2 wereviewsomeresultsconcerningthemultivariateEBestimators.InSection 4.3 weintroducethelimitedtranslationestimators.Section 4.4 evaluatestheirBayesriskundertheassumedpriorwhiletheirfrequentistriskisevaluatedinSection 4.5 .InSection 4.6 weundertakeasimulationstudytoevaluatetheeectivenessofthelimitedtranslationEBestimatorsand 56

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^Bi=Xi(A+)1(Xi)=XiB(Xi)=(IpB)Xi+B; whereB=(A+)1.However,^Bicannotbeusedassuchwheneitherorbothofthepriormeanandpriorvariance-covariancematrixareunknown.AnEBestimatorisobtainedbyobservingthatmarginallyXiiidNp(;A+),i=1;2;:::;n.Hence,theunknownisestimatedbyXn=n1Pni=1Xi.Also,lettingS=Pni=1(XiXn)(XiXn)T,theinverseoftheunknownmarginalvariancecovariancematrix(A+)1,isestimatedbyaS1,whereaisaknownconstantwhichwesetequaltonp2ifwewantanunbiasedestimatorof(A+)1.Thus,theresultingEBestimatorofiis ~EBi=XiaS1(XiXn)=Xi^B(XiXn)=(Ip^B)Xi+^BXn; where^B=aS1.TheEBestimatorshrinkstheMLestimatorofi,namelyXi,towardsthegrandmean,Xn,theMLestimatoroftheunknownpriormean,.Indoingso,theEBestimatorattainsalowerBayesriskthantheMLestimator,undertheassumedprior.However,itresultsinhighfrequentistriskwhentheiisfarfromthegrandmean.Inordertoquantifythetwoprecedingstatements,wecalculatetheBayesandfrequentistrisks,underthematrixlossfunctionL1,oftheEBestimator.ThefollowingTheoremprovidesanexpressionfortheBayesriskof~EBi,whichisdenotedbyr1(;~EBi)andcalculatedas 57

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Theproofisgiveninsection C.1 ofAppendixC.TheaboveBayesriskisminimalwhena=np2,andwiththischoiceofa,itbecomes whichincreaseswithpbutdecreaseswithn.Thisisintuitivelyobvioussinceaspincreasessodoesthenumberofparameterstobeestimated.Ontheotherhand,asnincreasessodoestheinformationavailablefortheestimationofthepriorparameters.NoticethatB=(A+)1ispositivedenite.Hence,itisclearthattheaboveBayesriskissmallerthantheBayesriskoftheMLestimatorXi,ofi,r1(;Xi)=.WenowturnourattentiontothefrequentistriskoftheEBestimator~EBiofi.ThisriskisdenotedbyR1(i;~EBi)anditiscalculatedbyaveragingthematrixlossfunctionL1overthesamplingdistributionofX,thatisR1(i;~EBi)=Ef(i~EBi)(i~EBi)Tg.Inordertoobtainanunbiasedestimatoroftheriskof~EBi,weusethemultivariateversionofStein'sidentityprovidedinthefollowingLemma. Stein ( 1981 )fortheunivariatecaseandthusomitted. 58

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UsingthemultivariateversionofStein'sidentity,weobtainanotherexpressionforthefrequentistriskoftheEBestimator.ThisexpressionisgiveninthefollowingTheorem. Theproofisdeferredtosection C.2 ofAppendixC.Now,let~EB=(~EB1)T;:::;(~EBn)TTandconsiderthetheloss Underthisloss, Inparticular,fora=np2,whichminimizestheabove,wehavethatR2(;~EB)=naE(S1),anditisclearthatR2(;~EB)
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4{9 ,toconsiderthecenterofthedataXn,tobeequallyrelevantfortheestimationofalli.InordertoavoidhighrisksassociatedwithEBestimatorscorrespondingtooutlyingobservationsXi,andpossiblyoutlyingcorrespondingrandomeectsi,wedeveloplimitedtranslationEBestimatorsthatserveascompromisebetweentheEBandtheMLestimators.SincetheMLestimatorshaveminimaxriskequaltoforalliweexpectthelimitedtranslationEBestimatorsalsotomaintainlowfrequentistriskandadditionallytomaintainlowBayesrisk. ^EBi=XiB(XiXn): Basedonthisestimator,somerobustestimators,namelythethelimitedtranslationEBestimators,areobtainedbymodifyingitinawaythatcontrolstheamountofshrinkageofXitowardsXn.Thisisdonebycontrollingin^EBithestandardizedversionofXiXn. ^LEBc;i=XiB(XiXn)c(jjD1 2(XiXn)jj2);wherec(u)=min(1;c=p Efron&Morris 1971 1972a ),cisaknownconstantandDvar(XiXn)=(11=n)(A+).Wemaynotethesimilaritybetweentherelevancefunction,c(u),of Efron&Morris ( 1971 1972a )withthefunction,hc(u),of Huber ( 1974 ).Theyareconnectedthroughtheequality,hc(u)=uc(u). 60

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~LEBc;i=Xi^B(XiXn)c(jjk1S1 2(XiXn)jj2); wherek1=(11=n)1 2a1 2.ItcanalsobewrittenasaweightedaverageoftheMLandEBestimatorssince ~LEBc;i=Xif1c(jjk1S1 2(XiXn)jj2)g+~EBic(jjk1S1 2(XiXn)jj2): ThelimitedtranslationEBestimatorfollowstheEBestimatorascloselyaspossiblesubjecttotheconstraintthatthedistanceoftheobservedXitotheobservedmeanXn,asmeasuredbyjjk1S1 2(XiXn)jj,doesnotexceedacertainvalue,csay.Whenthisdistancetakesonavaluebiggerthanc,therelevancefunctiontakesonavaluesmallerthanone,andfrom 4{12 weseethatthelimitedtranslationEBestimatorgivestheMLEbiggerweightattheexpenseoftheweightoftheEBrule.AsthedistanceofXitoXnincreasesthelessrelevanttheEBruleisconsideredasanestimatorofi.

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C.3 ofAppendixC.TheBayesriskof~LEBc;iisminimalfora=np2,inwhichcase Letting1sc=Ef1c(aW)g2theexpressionforr1(;~LEBc;i)becomes Thegeneralizedrelativesavingslossof~LEBc;iwithrespecttoXiisdenedas GRSL(~LEBc;i;Xi)=[r1(;Xi)r1(;~EBi)]1[r1(;~LEBc;i)r1(;~EBi)]: Thetermr1(;Xi)r1(;~EBi)isthesavings,inBayesrisksense,thatoccurwhenusingtheEBestimatorinsteadoftheMLE,whiler1(;~LEBc;i)r1(;~EBi)isthelossthatoccurswhenusing~LEBc;iinsteadoftheEBestimator.Hence,fora=np2,thegeneralizedrelativesavingslossof~LEBc;iwithrespecttoXiisiscalculatedasGRSL(~LEBc;i;Xi)=(1sc)Ip.Wenowgiveanexpressionfor1sc=Ef1c(aW)g2forthechoiceofrelevancefunctionc(u)=min(1;c=p 1sc=Ef1c(aW)g2=Ef1min(1;c=p 2(p+1 2)(n+1 2)1(p+2 2)1(n whereWi,i=0;1;2,havetheBeta((p+2i)=2;(np1)=2)distributionsrespectively.TheBayesriskofthelimitedtranslationEBestimatorisaweightedaverageoftheBayesrisksoftheEBestimatorandthatoftheMLE,theweightsbeingscand1screspectively.Thiscausesalossinthegeneralizedsavingsof(1sc)Ip.However,the 62

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4-1 showshow1scdecreasesascincreasesfortwovaluesofp=2;5andtwovaluesofn=10;30.Itisinterestingtoobservethatforgivenvaluesofcthesmallestvaluesof1scoccurforp=5andn=10whilethebiggestonesoccurforp=5andn=30.Anintuitiveinterpretationofthis,assumingcorrectlyspeciedpriors,wouldbethatwhennissmallcomparedtop,theuncertaintyassociatedwiththeEBestimatorisquitehigh.Thus,usingthelimitedtranslationEBestimatorinsteadoftheEBestimatordoesnotcausemuchlossinBayesrisk.Ontheotherhand,whennislargecomparedtopandbothnandparelarge,muchinformationislostbyusingtheMLestimatorinsteadoftheEBestimator,andthelimitedtranslationEBestimatorisindeedacompromisebetweentheMLandtheEBestimators.Anotherchoicefortherelevancefunctionwouldbe ThisrelevancefunctionreectstheideathattheEBruleisirrelevantforobservationsthatforwhichjjk1S1 2(XiXn)jj>c0wherec0>c,andthecorrespondingGRLSisgivenby 1sc;c0=Ef1c;c0(aW)g2=Ef1min(1;c=p 2I[(c2=a)
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2)(n+1 2)1(p+2 2)1(n whereWi;i=0;1;2;aresameasinEquation 4{17 2(XiXn)jj2)andtoshortennotationwewritecforc(jjk1S1 2(XiXn)jj2).Theriskof~LEBc;iiscalculatedas UsingthemultivariateversionofStein'sidentity,giveninLemma 4.2.2 ,weobtainthefollowingexpressionfortheriskofthelimitedtranslationestimator. 2(XiXn)jj2I[jjk1S1 2(XiXn)jj>c]o+2aEtrfS1(XiXn)(XiXn)Tg+n11S1c; Theproofisgiveninsection C.4 ofAppendixC.Thus,anunbiasedestimatorofR1(i;~LEBc;i)isgivenas ^R1(i;~LEBc;i)=+2ca2S1(XiXn)(XiXn)TS1+2actrfS1(XiXn)(XiXn)Tg1+n1S1

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2(XiXn)jj2I[jjk1S1 2(XiXn)jj>c]S1(XiXn)(XiXn)TS1: ThisquantitydoesnottakelargevaluesevenifXiisfarfromXn.ThisisbecauseofthepresenceofthefunctioncineachofthetermsthatdependonXiXn.WhenthedistancefromXitoXn,asmeasuredbyjjk1S1 2(XiXn)jj,takesonalargevalue,biggerthanc,c=c=jjk1S1 2(XiXn)jjtakesonasmallvalue.Wearethusprotectedfromlargefrequentistrisks,regardlessofhowwelltheassumedpriorresemblesthetruepriordistribution. NotethattheodiagonalelementsofA2weresochosentokeepthecorrelationsofi1andi2sameasthoseimpliedbyA1,cor(i1;i2)=0:2.Inthesecondscenario,weobtaintheifromthenormaldistributioniiid2N2(0;A3),whereA3=2644:50:90:94:5375,isthevariance-covariancematrixofthecontaminatednormaldistribution.Figure 4-2 showstheiobtainedfromthecontaminatedmodelaswellasthoseobtainedfromthenormaldistribution. 65

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Atthethirdstep,weestimateeachofthei,i=1;:::;n,ofthetwosets.Usingthedatathatcamefrombothf1andf2,wecalculatetheestimatorsXi;~EBiand~LEBc;i.Forthelimitedtranslationestimators,weconsiderthreevaluesfortheconstantc.Correspondingtogeneralizedrelativesavingsloss,1sc=:1thevalueofcisc=1:251,whilefor1sc=:05and1sc=:01thevaluesofcare1:517and2:028respectively.ThesecondandthirdstepsarerepeatedR=10;000times.Let^ibeanyestimatorofi.Then,thefrequentistriskof^i,R1(i;^i)=E(i^i)(i^i)T,isapproximatedby^R1(i;^i)=R1PRr=1(i^i;r)(i^i;r)T,where^i;ristheestimateofifromtherthrun.Sincethe^R1arematrices,wecalculatetheirtraces,ti=tr[^R1(i;^i)],andtheirdeterminants,di=det[^R1(i;^i)],asonenumbersummaries.Now,foreachofthetwosetsofiandforeachofthetwosamplingsdistributions,f1andf2,wesummarizethedistributionsoftheresultingtianddi,i=1;:::;30,byreportingtheminimumvalues(Q0),the25thpercentiles(Q0:25),themedians(Q0:50),the75thpercentiles(Q0:75),themaximumvalues(Q1),themeans(Mean)andthestandarddeviations(Stdev).TheresultsaredisplayedinTables 4-1 4-4 ,whereineachcelltherstentrydescribesthequantilesoftheti,whilethesecondonedescribesthequantilesofthedi.SomeinterestingissuesemergeoutofTable 4-1 .First,eveninthecasewheretheassumedpriorisnotveryclosetothetrueprior,theEBestimatorperformswell.Asfarastheaverages(Mean)oftianddiareconcerned,itdoesbetterthantheMLE.Onaverage,thetwomeasuresoffrequentistriskoftheMLEarereducedby11:843=1:998= 66

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4-2 ,weseethattheEBestimatorhasmaximumrisks(Q1)31:157=10:275=3:03and108:290=25:376=4:26timesbiggerthatthoseoftheMLE.However,theEBestimatorreducestheaverage(Mean)risksoftheMLEby17:084=9:989=29:08%and112:997=23:950=45:73%respectively.ThelimitedtranslationestimatorscompareveryfavorablytotheEBestimators.Inparticular,theestimator~LEB1:251;idecreasesthemaximumrisks(Q1)oftheEBestimatorby112:473=31:157=59:96%and135:406=108:290=67:30%respectively.Italsoreducestheaverage(Mean)risksoftheEBestimatorby16:319=7:084=10:80%and110:249=12:997=21:14%respectively.ContinuingtoTable 4-3 ,whichdisplaystheresultsforthesetofithatwasobtainedfromthenormalprior2andthef1samplingdistribution,weseethattheEBandthelimitedtranslationestimatorshaveverysimilarperformance.ThelimitedtranslationestimatorshaveslightlybiggerMeanrisksthantheEBestimatorsbuttheyslightlydecreasethemaximumrisks(Q1)oftheEBestimators.Also,comparedtotheMLE,thelimitedtranslationestimators,havesmallerMeanrisksbutbiggermaximumrisks. 67

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4-4 displaysthecomparisonoftheestimatorsforthecasewheretheiwereobtainedfromthe2prior,andtheXifromthef2samplingdistribution.Thiscomparisonisverysimilartotheonewehaveseenforthe2priorandthef1samplingdistribution.Again,theEBandthelimitedtranslationestimatorsperformsimilarly.TheirslightdierencesarethatthelimitedtranslationestimatorshavebiggerMeanrisksthantheEBestimatorsbuttheydecreasethemaximumrisks(Q1)oftheEBestimators.Also,thelimitedtranslationestimatorshavesmallerMeanrisksbutbiggermaximumrisksthanthatoftheMLE.Inordertostudytheeectofthesamplesize,wetooksamplesofsize20fromeachofthetwosetsofthirtyi.OurnewsetsofiaredisplayedinFigure 4-3 (a)and(b).Thesecondandthirdstepsofthisstudyweresameastheonesdescribedearlier.Theresults,showninTables 4-5 4-8 ,areverysimilartotheonesthatwehavealreadyseen.Itisthussafetoconcludethatthesamplesizedoesnotaectmuchtheperformanceoftheestimatorsunderexamination.Wemaynotethatforn=20thevaluesofccorrespondingto1sc=10%;5%and1%arec=1:195;1:443and1:910respectively. Figure4-1. Plotof1scasafunctionofc,forp=2;10andn=10;30. 68

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(b)Figure4-2. Theigeneratedfrom(a)thecontaminatedmodeland(b)thenormalmodel. 69

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Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=30. Stdev 1.987 1.995 2.011 2.055 1.998 0.020 0.925 0.948 0.956 0.967 1.015 0.958 0.020 ~EBi 1.687 1.703 1.724 3.708 1.843 0.458 0.657 0.690 0.701 0.722 2.584 0.824 0.406 ~LEB1:251;i 1.689 1.704 1.729 2.149 1.758 0.141 0.657 0.691 0.703 0.726 1.109 0.749 0.121 ~LEB1:517;i 1.687 1.703 1.725 2.218 1.761 0.158 0.657 0.690 0.701 0.722 1.176 0.752 0.136 ~LEB2:028;i 1.687 1.703 1.724 2.386 1.772 0.194 0.657 0.690 0.701 0.722 1.335 0.761 0.169 Table4-2. Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=30. Stdev 9.935 9.975 10.054 10.275 9.989 0.102 23.128 23.689 23.889 24.162 25.376 23.950 0.499 ~EBi 5.086 5.227 5.420 31.157 7.084 5.949 5.588 6.302 6.676 7.299 108.290 12.997 21.455 ~LEB1:251;i 5.362 5.562 5.840 12.473 6.319 2.016 6.021 6.981 7.522 8.461 35.406 10.249 7.304 ~LEB1:517;i 5.179 5.352 5.585 13.707 6.223 2.354 5.703 6.525 6.971 7.747 40.426 10.037 8.478 ~LEB2:028;i 5.092 5.238 5.436 16.809 6.323 3.067 5.594 6.316 6.694 7.342 52.304 10.422 11.043 Table4-3. Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=30. Stdev 1.987 1.995 2.011 2.055 1.998 0.020 0.925 0.948 0.956 0.967 1.015 0.958 0.020 ~EBi 1.612 1.704 1.805 2.577 1.767 0.228 0.564 0.632 0.706 0.793 1.322 0.749 0.172 ~LEB1:251;i 1.634 1.761 1.892 2.211 1.790 0.178 0.566 0.646 0.752 0.874 1.099 0.772 0.143 ~LEB1:517;i 1.617 1.722 1.871 2.322 1.776 0.196 0.564 0.635 0.721 0.845 1.173 0.760 0.156 ~LEB2:028;i 1.613 1.704 1.814 2.534 1.769 0.226 0.564 0.632 0.707 0.797 1.308 0.752 0.174 70

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Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=30. Stdev 9.941 9.994 10.081 10.236 10.013 0.098 23.108 23.730 24.082 24.361 25.150 24.089 0.474 ~EBi 4.607 5.376 6.748 13.407 6.210 2.260 3.511 5.106 6.703 9.910 24.874 8.642 5.150 ~LEB1:251;i 5.202 6.067 7.468 11.332 6.605 1.771 4.285 6.581 8.708 12.669 23.924 10.389 5.080 ~LEB1:517;i 4.874 5.731 7.213 12.041 6.412 1.984 3.794 5.749 7.689 11.515 24.564 9.612 5.282 ~LEB2:028;i 4.647 5.434 6.882 13.019 6.255 2.207 3.540 5.188 6.870 10.298 25.045 8.888 5.307 71

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(b)Figure4-3. Asampleofigeneratedfrom(a)thecontaminatedmodeland(b)thenormalmodel. 72

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Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=20. Stdev 1.987 2.002 2.014 2.025 2.000 0.018 0.922 0.945 0.960 0.970 0.992 0.959 0.017 ~EBi 1.665 1.683 1.708 2.825 1.838 0.378 0.659 0.676 0.689 0.710 1.691 0.822 0.324 ~LEB1:251;i 1.672 1.685 1.719 2.200 1.764 0.180 0.660 0.681 0.694 0.722 1.139 0.759 0.154 ~LEB1:517;i 1.665 1.683 1.710 2.287 1.769 0.207 0.659 0.677 0.690 0.712 1.212 0.764 0.176 ~LEB2:028;i 1.665 1.683 1.708 2.482 1.792 0.267 0.659 0.676 0.689 0.710 1.382 0.783 0.226 Table4-6. Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=20. Stdev 9.908 10.041 10.087 10.256 10.012 0.112 23.176 23.521 24.116 24.234 25.035 24.009 0.515 ~EBi 5.145 5.322 5.747 21.543 7.137 4.509 5.968 6.461 6.824 7.846 69.833 13.322 16.520 ~LEB1:251;i 5.479 5.740 6.200 12.124 6.665 2.254 6.729 7.365 7.996 9.196 31.778 11.490 8.317 ~LEB1:517;i 5.271 5.492 5.948 13.079 6.596 2.623 6.245 6.797 7.293 8.428 34.942 11.358 9.535 ~LEB2:028;i 5.157 5.340 5.773 14.821 6.705 3.231 5.995 6.496 6.872 7.919 42.474 11.777 11.572 Table4-7. Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=20. Stdev 1.997 2.003 2.016 2.029 2.006 0.013 0.945 0.956 0.965 0.976 0.986 0.965 0.011 ~EBi 1.744 1.799 1.898 2.182 1.835 0.146 0.656 0.737 0.780 0.842 1.055 0.807 0.115 ~LEB1:251;i 1.796 1.861 1.974 2.125 1.871 0.140 0.656 0.764 0.837 0.905 1.036 0.840 0.115 ~LEB1:517;i 1.763 1.825 1.939 2.180 1.856 0.152 0.656 0.744 0.802 0.874 1.069 0.826 0.123 ~LEB2:028;i 1.745 1.800 1.901 2.214 1.841 0.156 0.656 0.737 0.781 0.844 1.082 0.812 0.124 73

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Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=20. Stdev 9.920 9.940 9.988 10.057 9.958 0.053 23.472 23.678 23.791 24.057 24.242 23.829 0.232 ~EBi 5.964 6.588 7.305 10.600 6.925 1.638 5.455 8.235 10.149 11.821 19.664 11.022 4.305 ~LEB1:251;i 6.517 7.233 7.842 10.205 7.342 1.430 6.216 9.791 12.621 14.170 20.892 12.760 4.412 ~LEB1:517;i 6.242 6.947 7.641 10.451 7.164 1.563 5.717 8.975 11.504 13.123 20.733 12.011 4.555 ~LEB2:028;i 6.012 6.668 7.396 10.648 6.988 1.654 5.475 8.354 10.416 12.182 20.162 11.271 4.478 74

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5.1 .TheBayesriskoftheEBandlimitedtranslationEBestimatorsareevaluatedinsections 5.2 and 5.3 respectively.Sections 5.4 and 5.5 examinethefrequentistriskoftheestimators.Insection 5.6 weundertakeasimulationstudytofurtherevaluatethefrequentistriskperformance.Further,insection 5.7 weapplytheempiricalBayesandlimitedtranslationempiricalBayesestimatorsinordertoestimatetheaveragevitaminintakesofHIV-negativedrugabusers.TheproofsoftheresultsofthesechapteraregiveninAppendixD. wherej=1;2;:::;kandi=1;2;:::;n.Let=k1andXi=k1Pkj=1Xij.InordertoderiveBayesestimatorsforthei,notethatfori=1;2;:::;n, XijiindNp(i;);iiidNp(;A): Thus,whenalltheparametersareknown,theBayesestimatorofi,i=1;2;:::;n,withrespecttothematrixlossfunctionL1(i;a)=(ia)(ia)T,isgivenby ^Bi=Xi(A+)1(Xi)=(IpB)Xi+B=XiB(Xi); 75

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^EBi=(IpB)Xi+BX=XiB(XiX); whichisaweightedaverageoftheMLEofi,Xi,andthesamplemean,X.Thisestimator,incontrastwiththeMLE,usesinformationincludedinthewholedatasetandnotjustthedatacorrespondingtotheithindividualorpopulation.Additionally,wehavedenedthelimitedtranslationestimatorofmaximumtranslationcas ^LEBc;i=XiB(XiX)c(jj(11=n)1 2(A+)1 2(XiX)jj2); wherec(u)=min(1;c=p Theinverseoftheunknownmarginalvariance-covariancematrix,(A+)1,isestimatedbyaS1and,thevariance-covariancematrixofthesamplingdistribution,isestimated 76

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EBi=Xi~B(XiX)LEBc;i=Xi~B(XiX)c(jjk1S1 2(XiX)jj2); where~B=abk1VS1andk1=a1 2(11=n)1 2.Inthefollowingsectionswecomparethepropertiesoftheseestimators.WebeginbyconsideringtheirBayesriskperformance.Insubsequentsectionswealsostudytheirfrequentistrisksbybothderivingunbiasedestimatorsoftheserisksandbyundertakingasimulationstudy.Finally,weapplytheproposedinferentialprocedureinordertoestimatethelongtermaveragevitaminintakesofHIV-negativedrugabusers. Theproofisgiveninsection D.1 ofAppendixD.Considernowthequadraticlossfunction,L2,givenby Forxeda=np2,thechoiceofwhichresultsinanunbiasedestimatorofthemarginalvariance-covariancematrix,asdiscussedinthepreviouschapter,theBayesriskoftheEBestimator,undertheL2lossfunctionisminimizedforb=fn(k1)+p+1g1.Withthis 77

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Clearly,r2(;EBi)
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ItisclearthattheproposedestimatorhassmallerBayesriskthantheMLE.However,ithasslightlybiggerBayesriskthantheregularEBestimator.Inordertoquantifythelaststatementofthepreviousparagraph,weusetheconceptoftherelativesavingslossofLEBc;iwithrespecttoXi,denedas RSL(LEBc;i;Xi)=[r2(;Xi)r2(;EBi)]1[r2(;LEBc;i)r2(;EBi)]: Thetermr2(;Xi)r2(;EBi)isthesavings,inBayesrisksense,thatoccurwhenusingtheEBestimatorinsteadoftheMLE,whiler2(;LEBc;i)r2(;EBi)isthelossthatoccurswhenusingLEBc;iinsteadoftheEBestimator.Itcanbeshown,usingEquations 5{11 and 5{14 ,thatRSL(LEBc;i;Xi)=12E(c)+E(2c)=E(1c)2=1sc.InFigure 5-1 weplotRLSagainstcanditcanbeseenthatRLSisadecreasingconvexfunctionofc.WecanthuschoosecbydecidingbywhatpercentageisworthincreasingtheBayesriskoftheEBestimatorinordertoreceiveprotectionagainlargefrequentistrisks.TheconstantccanbesochosenthatRLS=0:05or0:01.Bysacricing5%or1%oftheBayesrisk,wereceiveconsiderableprotectionagainstlargefrequentistrisks.Inordertomakethelatterpointclear,wewillexamineandcomparethefrequentistrisksofEBiandLEBc;iinthesubsequentsections.Finally,wegiveanotherexpressionforRLS RLS=P(W0>a1c2)+c2(n1)(ap)1P(W2>a1c2)+ca1 2(p+1 2)1(p+2 2)(n+1 2)1(n whereWi,i=0;1;2,havetheBeta((p+2i)=2;(np1)=2)distributionsrespectively. 79

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4.2.2 .First,thefrequentistriskisofEBcalculatedas withtheexpectationbeingtakenwithrespecttothesamplingdistributionoftheXij,i=1;:::;n,j=1;:::;k,andassumingthat=(1)T;:::;(n)TTisxed.UsingStein'sidentityweobtainanotherexpressionfortherisk. Theproofisgiveninsection D.3 ofAppendixD.LetEB=n(EB1)T;:::;(EBn)ToTandconsiderthelossobtainedbyaveragingtheindividualL2losses.Theaveragequadraticloss,usingtheresultofEquation 5{18 ,andfora=np2andb=fn(k1)+p+1g1,canbeshowntobeequalto R2(;EB)=n1nXi=1Ef(iEBi)T1(iEBi)g=n1nXi=1trf1R1(i;EBi)g=p(np2)2(k1) whichforallislessthantheriskoftheMLE.ThelatterisequaltoR2(;X)=panditisclearlybiggerthantheriskinEquation 5{19 .Inotherwords,theEBestimatordominatestheMLestimator,andthusitisaminimaxestimator.Aninterestingapproach 80

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Efron&Morris ( 1972b ).ReturningtothegeneralresultofTheorem 5.4.1 ,wecanobtainanunbiasedestimatorofthefrequentistriskofEBiunderthequadraticlossL2.Thisestimatorisgivenby ^R2(i;EBi)=p+pa2b3nk1(k1)trS1(XiX)(XiX)TS1V+ab2nk1(k1)b[abfn(k1)+1g+2]VS1(XiX)(XiX)TS1+2ab2nk1(k1)htrS1(XiX)(XiX)T1+1=niVS1: TheabovequantitywillbelargewhenXiisfarfromX.WhenkislargeandtheriskassociatedwiththeMLestimatorsissmall,observinganoutlyingXi,mightbeanindicationthatthecorrespondingiisfarfromtherestofthe's.Underthisscenario,itisnotverywisetoshrinkbyalottheMLEofitowardsthecenterofthedata.IntuitionaswellasEquation 5{20 suggestthat,insuchacase,theriskattachedtoEBiisquitehigh. 2(XiX)jj2).First,theriskofLEBc;iiscalculatedas AnotherexpressionoftheaboveriskisgiveninthefollowingTheorem. 81

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+2abn(k1)EhtrfS1(XiX)(XiX)TgS1ci2abn(k1)(11=n)E(S1c)+2abn(k1)EhS1(XiX)(XiX)TS1cn(11=n)jjS1 2(XiX)jj2oI[jjk1S1 2(XiX)jj>c]i: TheproofisdeferredtotheAppendixD,section D.4 .BasedonthegeneralresultofTheorem 5.5.1 ,weobtainanunbiasedestimatoroftheriskofLEBc;iunderthequadraticlossL2.Thisestimatorisgivenby ^R2(i;LEBc;i)=p+pa2b3nk1(k1)2ctrS1(XiX)(XiX)TS1V+habfn(k1)+1gc+2n(11=n)jjS1 2(XiX)jj2oI[jjk1S1 2(XiX)jj>c]iab2nk1(k1)cVS1(XiX)(XiX)TS1+2ab2nk1(k1)ctrS1(XiX)(XiX)T1+1=niVS1: TheaboveestimatoroftherisksuggeststhatLEBc;idoesnotallowforlargefrequentistrisks.ThisisbecauseofthepresenceofthefunctioncineachofthetermsthatdependonXiX.WhenthedistancefromXitoXbecomeslargerthanc,thenc=c=jjk1S1 2(XiX)jjtakesonasmallvaluenotallowingtherisktobecomelarge. 82

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4.6 .Itconsistsofthreestepswhichwebrieydescribe.Attherststepweobtainvaluesforthei,i=1;:::;n,andkeepthemxed.Wetaken=30andthedimensionoftheitobep=2.Themethodofobtainingiisexactlythesameastheonedescribedinsection 4.6 .WemayrecallthatthetwosetsofiareshowninFigure 4-2 .AtthesecondstepofthesimulationstudywegeneratetheobservationsXij,i=1;:::;n,j=1;:::;k,wherekwasselectedtobek=3.Foreachofthetwosetsofi,wegeneratetheXijrstlyasXijjiindf1N2(i;1)andsecondlyasXijjiindf2N2(i;2),where1and2aregivenin 4{24 .Atthethirdstep,weestimateeachofthei,i=1;:::;n,ofthetwosetsusingtheobservationsthatcamefrombothf1andf2.WecalculatetheestimatorsXi;EBiandLEBc;i.Forthelimitedtranslationestimators,weconsiderthreevaluesfortheconstantc.Correspondingto1sc=:1thevalueofcisc=1:251,whilefor1sc=:05and1sc=:01thevaluesofcare1:517and2:028respectively.ThesecondandthirdstepsarerepeatedR=10;000times.Let^ibeanyestimatorofi.Then,thefrequentistriskof^i,R1(i;^i)=Ef(i^i)(i^i)Tg,isapproximatedby^R1(i;^i)=R1PRr=1(i^i;r)(i^i;r)T,where^i;ristheestimateofifromtherthrun.Sincethe^R1arematrices,wecalculatetheirtraces,ti=tr[^R1(i;^i)],andtheirdeterminants,di=det[^R1(i;^i)],asonenumbersummaries.Foreachofthetwosetsofiandforeachofthetwosamplingsdistributions,f1andf2,wesummarizethedistributionsoftheresultingtianddi,i=1;:::;30,byreportingtheminimumvalues(Q0),the25thpercentiles(Q0:25),themedians(Q0:50),the75thpercentiles(Q0:75),themaximumvalues(Q1),themeans(Mean)andthestandarddeviations(Stdev).TheresultsaredisplayedinTables 5-1 5-4 ,whereineachcellthetwoentriesdescribethequantilesofthetianddirespectively. 83

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5-1 ,werstnoticethattheentriesoftherowthatcorrespondstotheMLEareveryclosetothetheoreticalvalues,tr(k11)=0:667anddet(k11)=0:107,withverysmallvariability.Now,eveninthecasewheretheassumedpriorisnotveryclosetothetrueprior,theEBestimatorperformswell.Theaverages(Mean)oftianddifortheEBestimatoraresmallerthanthoseoftheMLE.Onaverage,thetwomeasuresoffrequentistriskoftheMLEarereducedby10:649=0:666=2:55%and10:101=0:107=5:94%respectivelybytheEBestimator.NotonlythemeansbutalltheentriesofcolumnsQ0Q0:75correspondingtoEBiaresmallerthantheonescorrespondingtoXi.ThebadpropertyoftheEBestimatoristhatitcanresultinlargefrequentistrisksforobservationthatarefarfromX.ThisbecomesobviousbyobservingtheentriesofQ1.ThetwoentriesinQ1fortheestimatorEBiare0:863=0:680=1:27and0:173=0:110=1:57timesbiggerthanthoseoftheminimaxestimator,Xi.WenowcomparethethreelimitedtranslationestimatorstotheEBestimator.Firstly,weobservethattheentriesincolumnsQ0Q0:75areidentical.Thelimitedtranslationestimators,however,havesmallermaximum(Q1)risksthantheEBestimator,sincetheybecomeclosertotheminimaxestimatorsforoutlyingobservations.Theythushavesmalleraverage(Mean)riskthanthatoftheusualEBestimator.Forinstance,LEB1:251;ireducesthemaximumrisksoftheEBestimatorby10:686=0:863=22:05%and10:113=0:173=34:68%.InTable 5-2 ,weseethattheEBestimatorhasmaximumrisks(Q1)7:282=3:332=2:19and8:953=2:780=3:22timesbiggerthatthoseoftheMLE.However,itreducestheaverage(Mean)risksoftheMLEby12:938=3:332=11:82%and12:127=2:666=20:22%respectively.ThelimitedtranslationestimatorscompareveryfavorablytotheEBestimators.Inparticular,theestimatorLEB1:251;idecreasesthemaximumrisks(Q1)oftheEBestimatorby13:746=7:282=48:56%and13:300=8:953=63:14%respectively.Italsoreducestheaverage(Mean)risksoftheEBestimatorby12:752=2:938=6:33%and11:852=2:127=12:92%respectively. 84

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5-3 and 5-4 displaythecomparisonoftheestimators,forthecasewheretheiwereobtainedfromthenormalprior2,andtheXifromthef1andf2samplingdistributionsrespectively.InbothTablesweseethattheEBandthelimitedtranslationestimatorsperformverysimilarly.Theirslightdierencesarethatthelimitedtranslationestimatorshavebiggeraveragerisks(Mean)thantheEBestimatorsbuttheydecreasethemaximumrisks(Q1)oftheEBestimators.Further,thelimitedtranslationestimatorshavesmallerMeanrisksbutbiggermaximumrisksthanthatoftheMLE.Inordertostudytheeectofthesamplesize,wetooksamplesofsize20fromeachofthetwosetsofthirtyithatwereobtainedintherststepofthesimulationstudy,seeFigure 4-3 (a)and(b).Thesecondandthirdstepsofthisstudyweresameastheonesdescribedearlier.Notethatforn=20,thevaluesofcthatcorrespondtoRLS=10%;5%and1%arec=1:195;1:443and1:910respectively.Theresultswereverysimilartotheonesthatwehavealreadyseen. 85

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Box&Cox ( 1964 ),transformation.Itturnsoutthatthevaluesofi,i=1;2,forthetransformationare1=0:10and2=0:35.Theaverageintakesofthetwovitaminsofthen=54subjects,afterthetransformation,aredisplayedinFigure 5-2 ,anditisclearthatevenafterthetransformationtheassumptionsofnormalityarenotexactlymet.Forthisreason,arobustprocedure,likethelimitedtranslationestimators,wouldbemoreappropriatethantheregularEBestimators.LetXij1denotetheintakeofvitaminAofpersoniindayj,and,likewise,Xij2theintakeofvitaminB1ofpersoniindayj.Further,Xij=(Xij1;Xij2)Tistheresponsevectorofpersoniindayj.Additionally,i1andi2denotetheaveragedailyintakeofvitaminAandB1,respectively,ofpersoni.Thevectori=(i1;i2)Tisaccordinglydened.Now,theEBandlimitedtranslationEBestimatorsarederivedbasedontheassumedmodel whichcanequivalentlybewrittenasamixedlinearmodelsince, Table 5.7 displaystheestimatedlongtermaverageintakesofthetwovitaminsforthersttenpatientsinoursample.TheestimateswereobtainedusingtheML,EBandlimitedtranslationEBestimators.TherstcolumnundereachheadingreferstotheestimatedvitaminAintakeswhilethesecondonereferstotheintakesofvitaminB1.NotethattheaveragevitaminAandB1intakes,aftertransformation,areX=(9:096;0:721)Twhile(n1)1S,theestimatedmarginalvariance-covariancematrixis 86

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5.7 ,weseethattheEBestimatorpoolstheMLestimatestowardsthegrandmean.ForthoseMLestimatesthatareclosetothegrandmean,theEBandthelimitedtranslationEBestimatesareidenticalwhileforthosethatarefarfromthegrandmean,thelimitedtranslationestimatesaresomewherebetweentheMLandEBestimates.Similarly,inFigure 5-3 ,weseetheML,EBandlimitedtranslationEBestimatesforalln=54subjects. Figure5-1. PlotofRLS=Ef1c(aW)g2asfunctionofc,forp=2;5andn=10;30. 87

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Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=30. Stdev Xi 0.664 0.666 0.669 0.680 0.666 0.006 0.103 0.106 0.106 0.108 0.110 0.107 0.002 EBi 0.629 0.634 0.641 0.863 0.649 0.055 0.092 0.095 0.097 0.099 0.173 0.101 0.017 LEB1:251;i 0.629 0.634 0.641 0.686 0.638 0.017 0.092 0.095 0.097 0.099 0.113 0.098 0.005 LEB1:517;i 0.629 0.634 0.641 0.695 0.639 0.019 0.092 0.095 0.097 0.099 0.115 0.098 0.006 LEB2:028;i 0.629 0.634 0.641 0.716 0.640 0.023 0.092 0.095 0.097 0.099 0.121 0.099 0.007 Table5-2. Comparisonoftherisksoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=30. Stdev Xi 3.314 3.328 3.353 3.401 3.332 0.030 2.584 2.632 2.659 2.699 2.780 2.666 0.048 EBi 2.568 2.610 2.652 7.282 2.938 1.073 1.566 1.602 1.660 1.724 8.953 2.127 1.583 LEB1:251;i 2.577 2.622 2.680 3.746 2.752 0.336 1.574 1.610 1.679 1.764 3.300 1.852 0.469 LEB1:517;i 2.570 2.612 2.656 3.926 2.753 0.380 1.567 1.603 1.664 1.729 3.563 1.855 0.531 LEB2:028;i 2.568 2.610 2.652 4.367 2.778 0.474 1.566 1.602 1.660 1.724 4.200 1.889 0.669 Table5-3. Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=30. Stdev Xi 0.660 0.666 0.672 0.690 0.666 0.009 0.101 0.104 0.106 0.108 0.114 0.106 0.003 EBi 0.619 0.632 0.650 0.745 0.640 0.034 0.085 0.092 0.096 0.102 0.125 0.098 0.009 LEB1:251;i 0.621 0.636 0.666 0.696 0.643 0.029 0.085 0.093 0.097 0.106 0.116 0.099 0.008 LEB1:517;i 0.619 0.633 0.659 0.708 0.641 0.030 0.085 0.092 0.096 0.104 0.118 0.099 0.009 LEB2:028;i 0.619 0.632 0.651 0.740 0.640 0.034 0.085 0.092 0.096 0.102 0.124 0.099 0.010 88

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Comparisonoftherisksoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=30. Stdev Xi 3.314 3.329 3.358 3.418 3.338 0.040 2.518 2.637 2.661 2.706 2.806 2.674 0.061 EBi 2.406 2.628 2.902 4.458 2.772 0.506 1.205 1.412 1.668 1.992 3.721 1.838 0.594 LEB1:251;i 2.471 2.758 3.075 3.779 2.832 0.403 1.223 1.486 1.827 2.264 3.099 1.933 0.507 LEB1:517;i 2.425 2.682 3.013 4.004 2.800 0.445 1.208 1.433 1.732 2.165 3.328 1.887 0.551 LEB2:028;i 2.407 2.632 2.919 4.375 2.779 0.501 1.205 1.413 1.672 2.022 3.678 1.852 0.601 Table5-5. Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef1samplingdistribution,n=20. Stdev Xi 0.660 0.666 0.671 0.678 0.666 0.007 0.102 0.105 0.106 0.108 0.109 0.106 0.002 EBi 0.626 0.633 0.640 0.750 0.647 0.042 0.093 0.095 0.096 0.099 0.133 0.101 0.012 LEB1:251;i 0.626 0.633 0.641 0.682 0.638 0.018 0.093 0.095 0.096 0.099 0.111 0.098 0.005 LEB1:517;i 0.626 0.633 0.640 0.693 0.638 0.021 0.093 0.095 0.096 0.099 0.114 0.098 0.006 LEB2:028;i 0.626 0.633 0.640 0.718 0.641 0.028 0.093 0.095 0.096 0.099 0.120 0.099 0.008 Table5-6. Comparisonoftheriskoftheestimatorsunderthecontaminatedmodelandthef2samplingdistribution,n=20. Stdev Xi 3.309 3.324 3.346 3.374 3.322 0.032 2.527 2.628 2.656 2.684 2.722 2.648 0.048 EBi 2.547 2.575 2.680 5.275 2.928 0.837 1.499 1.594 1.627 1.744 5.606 2.119 1.190 LEB1:251;i 2.559 2.599 2.725 3.759 2.781 0.402 1.512 1.610 1.653 1.807 3.286 1.904 0.558 LEB1:517;i 2.549 2.580 2.693 3.951 2.787 0.465 1.501 1.596 1.632 1.762 3.557 1.915 0.645 LEB2:028;i 2.547 2.575 2.6817 4.388 2.831 0.590 1.499 1.594 1.627 1.745 4.165 1.977 0.819 89

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Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef1samplingdistribution,n=20. Stdev Xi 0.661 0.663 0.665 0.680 0.665 0.006 0.103 0.105 0.106 0.107 0.111 0.106 0.002 EBi 0.633 0.643 0.653 0.683 0.646 0.020 0.094 0.096 0.100 0.102 0.112 0.100 0.006 LEB1:251;i 0.636 0.653 0.662 0.694 0.651 0.020 0.094 0.097 0.102 0.105 0.113 0.102 0.006 LEB1:517;i 0.633 0.646 0.657 0.695 0.648 0.021 0.094 0.096 0.100 0.103 0.113 0.101 0.006 LEB2:028;i 0.633 0.643 0.654 0.688 0.647 0.021 0.094 0.096 0.100 0.102 0.114 0.101 0.006 Table5-8. Comparisonoftheriskoftheestimatorsunderthenormalmodelandthef2samplingdistribution,n=20. Stdev Xi 3.302 3.324 3.349 3.380 3.327 0.035 2.530 2.623 2.662 2.694 2.758 2.660 0.056 EBi 2.747 2.850 3.015 3.668 2.917 0.321 1.477 1.794 2.001 2.147 2.884 2.043 0.401 LEB1:251;i 2.838 2.989 3.156 3.562 2.995 0.301 1.487 1.900 2.205 2.339 2.845 2.157 0.394 LEB1:517;i 2.782 2.918 3.096 3.660 2.964 0.327 1.478 1.833 2.103 2.252 2.932 2.109 0.421 LEB2:028;i 2.750 2.856 3.025 3.714 2.930 0.337 1.477 1.797 2.010 2.160 2.951 2.060 0.425 90

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AverageintakesofvitaminsAandB1. Table5-9. MLE,EB,LTEstimates MLE EB LT 9.143 -0.171 8.451 0.086 8.760 -0.029 10.415 1.059 9.973 1.017 9.973 1.017 10.774 1.215 10.260 1.143 10.260 1.143 8.134 0.759 8.670 0.708 8.670 0.708 9.798 1.038 9.665 0.976 9.665 0.976 5.992 -0.521 6.700 -0.295 6.395 -0.392 11.537 2.178 11.340 1.862 11.438 2.018 9.319 1.430 9.732 1.236 9.638 1.280 10.412 1.054 9.968 1.013 9.968 1.013 8.890 0.518 8.847 0.567 8.847 0.567 91

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EstimatedintakesofvitaminsAandB1. 92

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Efron&Morris ( 1971 1972a ).ThemultivariatelimitedtranslationBayesestimatorsserveasacompromisebetweentheBayesandthemaximumlikelihoodestimators.WehavedemonstratedtheusefulnessofsuchestimatorsovertheusualBayesestimators,inaBayesrisksense,undermisspeciedpriors.Fromthecriteriaoffrequentistrisks,wehavedemonstratedtheusefulnessofsuchestimators,whentheyareusedforestimatingparameterswhichdepartwidelyfromtheassumedpriormeans.Additionally,wedevelopedmultivariatelimitedtranslationempiricalBayesestimatorsofthenormalmeanvectorwhichserveasacompromisebetweentheempiricalBayesestimatorsandthemaximumlikelihoodestimators.Weexaminedthepropertiesofsuchestimatorsanddemonstratedtheirusefulnessfromthefrequentistriskscriteria,whenthereiswidedepartureofanindividualobservationfromthegrandaverage.Futureworkwilldevelopsimilarestimatorsusing,however,shrinkageestimators,asthosesuggestedby Efron&Morris ( 1976 ),fortheunknownvariance-covariancematrices.Intheprocess,wewilladdressdirectlyestimationofthevariance-covariancematrixofamultivariatenormaldistributioninaverygeneralsetup. 93

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2.4.1 and 2.5.1 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=()hEf[2p+2()]dI[2p+2()b]gEf[2p()]dI[2p()b]gi: 2(Y)jj2b]e1 2jj(a)1 2(Y)jj2 2(Y)jj2d(2)p 2dY=Efjj(a)1 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=Ef[2p()]dI[2p()b]g=1Xk=0ek WenowdierentiatebothsidesofEquation A{2 withrespectto.Firstnotethat @(Y)T(a)1(Y)=2(a)1(Y): Hence, 2(Y)jj2b](a)1(Y)e1 2jj(a)1 2(Y)jj2 2(Y)jj2d(2)p 2dY=(a)1EfY 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=1Xk=0(k!)1Ef(2p+2k)dI[2p+2kb]g@(ek) 94

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@ek+ekk1@ @=(a)1()ek1(k): CombiningEquations A{4 and A{5 weobtain 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=()1Xk=0ek1(k) ThiscompletestheproofofLemma A.1.1 A.1.1 .Then 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=()()ThEf[2p()]dI[2p()b]g+Ef[2p+4()]dI[2p+4()b]g2Ef[2p+2()]dI[2p+2()b]gi+aEf[2p+2()]dI[2p+2()b]g: A{2 withrespectto.Notethat 95

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(a)1Ef(Y)(Y)T 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g(a)1(a)1Efjj(a)1 2(Y)jj2dI[jj(a)1 2(Y)jj2b]g=1Xk=0(k!)1Ef(2p+2k)dI[2p+2kb]g@2(ek) where @Tf(a)1()ek1(k)g=(a)1ek1(k)+(a)1()()T(a)1ek2f2+k(k1)2kg: SubstitutingthelastexpressioninEquation A{9 ,weobtain CombiningEquations A{9 and A{11 andcollectingtermsweobtainLemma A.1.2 A.1.1 and A.1.2 holdevenifwechangetheinequalitiesfrombto>bwithobviousmodications. 96

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2.4.1 2(X)jj2c2(1+g)]e1 2jj1 2(X)jj2 2(X)jj2d(2)p 2dX=EfQd1I[Q1c2(1+g)]g=1Xk=0ek whichisthesameasEquation A{2 with SinceEquation A{2 wasthebasisofprovingLemmas A.1.1 and A.1.2 ,theresultsofthesetwoLemmasholdforthespecialcasedenedbytheEquationsin A{13 .Thisgivesusthefollowingtwoequalities 2(X)jj2dI[jj1 2(X)jj2c2(1+g)]g=()hEf[2p+2()]dI[2p+2()c2(1+g)]gEf[2p()]dI[2p()c2(1+g)]gi; and 2(X)jj2dI[jj1 2(X)jj2c2(1+g)]g=()()ThEf[2p()]dI[2p()c2(1+g)]g+Ef[2p+4()]dI[2p+4()c2(1+g)]g2Ef[2p+2()]dI[2p+2()c2(1+g)]gi+Ef[2p+2()]dI[2p+2()c2(1+g)]g: Also,notethatforanyk kc(jj(1+g)1 21 2(X)jj2)=kcf(1+g)1Q1g)= 97

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Wenowwrite Becauseoftheformofthefunctionc(:),weneedtocalculate and WethusapplyEquation A{14 withd=0toobtain andEquation A{15 ,againwithd=0,toobtain 98

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Furthermore,theformofthefunctionc(:),giveninEquation A{16 ,andEquation A{17 indicatethatwealsoneedtondanexpressionfor 21I[Q1>c2(1+g)]=2Ef(X)(X)TQ1 21I[Q1>c2(1+g)]g+Ef(X)()T+()(X)TgQ1 21I[Q1>c2(1+g)]: TheresultsofEquations A{14 and A{15 withd=1=2andreversedinequalitiesshowthat 2I[2p+4()>c2(1+g)]gEf[2p+2()]1 2I[2p+2()>c2(1+g)]gi+2Ef[2p+2()]1 2I[2p+2()>c2(1+g)]g Finally,werewrite AgaintheresultsofEquations A{14 and A{15 withd=1andreversedinequalitiesshowthat 99

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A{16 A{25 andcollectingthecoecientsofand()()Tseparately,theresultfollows. 2.4.2 2.4.1 .Thatis,wecalculateR2(;^LBc)=tr[1R1(;^LBc)].Theresultingexpressioncanbesimpliedbymakinguseofthetwoequalitiesthatfollow.First, 2(p+2+2k 2(p+4+2k 2(p+2+2k k!(p+2k 2(p+2+2k 2(p+2+2k andsimilarly 2I[2p+2()>c2(1+g)]g+2Ef[2p+4()]1 2I[2p+4()>c2(1+g)]g=p1Xk=0ek 100

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k!(p+1+2k (p+2k 21ex 2I[2p()>c2(1+g)]g: TheproofofCorollary 2.4.2 isnowcomplete. 2.5.1 Also,marginallyXNp(;(1+g))sothatQ22p(),where=B()T1()=2.Theriskunderpriorofthelimitedtranslationestimatoriscalculatedas withthelastequalityfollowingfromthefactthatEf(^B)(^B^LBc)Tg=0.Wealsohavethatr1(;^B)=(1B),andweneedonlytocalculateEf(^B^LBc)(^B^LBc)Tg.Notethat^B^LBc=B(X)+B(X)c(BQ1).Hence, 101

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(X)(X)T=(X)(X)T+()(X)T and (X)(X)T=(X)(X)T+()()T+(X)()T+()(X)T; itfollowsfromEquation A{30 andsomesimplication Foranyk>0,wewrite andthetwocomponentsofthisexpectationsarecalculatedbyapplyingLemma A.1.2 andtheremarkfollowingitwithY=X,a=1+g,b=c2(1+g)(1+g)1,=,=,andd=0andk=2respectively.Wealsohavethat 102

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A.1.1 andtheremarkfollowingitwithY=X,a=1+g,b=c2(1+g)(1+g)1,=,=,andd=0andk=2respectively.Finally,wecollectingthecoecients()()Tandseparatelyandtheresultfollows. 2.5.2 A{26 and A{27 103

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3.4.1 2(XiXn)jj2)=kcfB(1n1)Qg=I[(1n1)Qc2(1+g)]+ckBk Wewrite Now, 2Ef(XiXn)(in)g(XiXn)Tf(11=n)1Qg1 2I[(11=n)1Q>c2(1+g)]; andtherstofthetwoexpectationsinthelastthreelinesoftheaboveequationiscalculatedbyapplyingLemma A.1.2 withY=XiXn,=in,a=11=n, 104

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andthesetwoexpectationsarecalculatedusingLemma A.1.2 exactlyaswedidinEquation B{3 ,withtheonlydierencebeingthatinthesecondofthetwoexpectationsoftheaboveequationwesetd=1insteadofd=1=2.TheresultfollowsfromcombiningEquations B{2 B{4 ,andcollectingthecoecientsofand(in)(in)Tseparately. 3.5.1 Now, ^EBi^LEBc;i=(XiXn)fBc(jjD1 2(XiXn)jj2)Bg: Therefore 2(XiXn)jj2)Bg2]: 105

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2(B)1 21 2Z,whereZNp(0;Ip).AlsothatjjD1 2(XiXn)jj2=(B=B)jjZjj2.Usingthelastequality,itiseasytoshowthatc(jjD1 2(XiXn)jj2)=c(jjZjj2)wherec=c(B=B)1 2.ReturningtoEquation B{7 ,wewriteMas 2E[ZZTfBc(jjZ)jj2)Bg2]1 2; anditfollowsfromEquation 3{22 that whereU2p+2.CombiningEquations B{5 and B{9 ,weobtaintheresult. 106

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4.2.1 Next,with^EBi=(IpB)Xi+BXn,wewrite Itiseasytoshowthat andalsothat withthelastequalityfollowingfromtheindependenceofXnand(XiXn;S).FindinganexpressionforEf(^EBi~EBi)(^EBi~EBi)Tgcompletesthetask.First, ^EBi~EBi=^B(XiXn)B(XiXn)=(^BB)(XiXn); andthus 107

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Also,E(S)=(n1)(A+)andE(S1)=(np2)1(A+)1.Thus,byexpansion, TheproofiscompletedbycombiningEquations C{1 C{4 and C{8 4.2.3 WewriteX(i)=(XT1;:::;XTi1;XTi+1;:::;XTn)TandusingtheresultofLemma 4.2.2 ,wecanseethat whichwhencombinedwithEquation C{9 givesus Next,observethat 108

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Also,itistruethat whereXijisthejthelementofvectorXi.SinceS=Pnk=1XkXTknXnXTn, FromEquations C{14 and C{15 wherefjisthejthcolumnofmatrixS1=(f1;:::;fp).Now,fromEquation C{16 ,weseethat (@S1=@Xij)(XiXn)=S1(XiXn)(XiXn)TfjtrfS1(XiXn)(XiXn)Tgfj; andthus (@S1=@Xi)(XiXn)=S1(XiXn)(XiXn)TS1trfS1(XiXn)(XiXn)TgS1: TheresultofEquation C{18 ,alongwithEquations C{12 and C{13 ,showsthat 109

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4.2.3 followsfromcombiningEquations C{11 and C{19 4.4.1 2(X1++Xn)=p 2(X1X2)...Yn=fn(n1)g1 2(X1++Xn1(n1)Xn)=(11=n)1 2(XnXn): Then, 2Yn; andYiiidNp(0;A+),i=2;:::;n.Accordingly, ~LEBc;n=Xn+(11=n)1 2^BYnc(jja1 2S1 2Ynjj2); ~EBn=Xn+(11=n)1 2^BYn: Nextwecalculater1(;~LEBc;n), TheBayesriskof~LEBc;nisnowwrittenas 110

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~EBn~LEBc;n=^B(XnXn)fc(jjk1S1 2(XnXn)jj2)1g; wherek1=(11=n)1 2a1 2.Also,writingd=asequalindistribution 2(nXi=2YiYTi)1Yn=(11=n)1 2(n1Xi=2YiYTi+YnYTn)1Ynd=(11=n)1 2(A+)1 2(W1+ZZT)1Z; whereW1W(Ip;n2;)independentlyofZN(0;Ip).Further, 2(XnXn)jj2=(XnXn)TS1(XnXn)d=(11=n)ZT(W1+ZZT)1Z=(11=n)jj(W1+ZZT)1 2Zjj2: FromEquations C{26 C{28 ,itfollowsthat 2Eh(W1+ZZT)1ZZT(W1+ZZT)1fc(ajj(W1+ZZT)1 2Zjj2)1g2i(A+)1 2: Wenowcontinuewiththecalculationof 2Zjj2)1g2i=M1;02M2;0+M3;0; where,fork=1;2;3andl=0;1,Mk;laredenedas 2Zjj2)o: Forthetimebeing,weneedonlyMk;0,k=1,2,3,butwewillneedMk;1,k=1,2,inordertocalculatethecrossproduct,Ef(n~EBn)(~EBn~LEBc;n)Tg.Now,letU= 111

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2Z.ThenthematricesUUTandW1+ZZTareindependentlydistributed( Srivastava&Khatri ( 1979 ),p.95).Thisindependenceresultallowsustosimplifymatters.WerstrewriteMk;las 2UUT(W1+ZZT)1 2k1c(ajjUjj2)o: NoticethatjjUjj2=tr(UUT).Thus,alternatively,Mk;liswrittenas 2Hk(W1+ZZT)1 2g; whereHk=EfUUTk1c(ajjUjj2)g.Now,thedensityoftherandomvectorUisgivenby 2I[uTu1]=c(1uTu)np3 2I[uTu1]; ( Srivastava&Khatri ( 1979 ),p.95),whereI[:]istheindicatorfunctionandcisthenormalizingconstant.Itfollowsthatfori6=j, TheaboveequalityholdstrueifandonlyifEfUiUjk1c(ajjUjj2)g=0,i6=j.Further,wehavethat 2du1:::dup 2du1:::dup: Inordertoevaluatetheaboveintegralweconsiderthepolartransformation 112

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EfU2ik1c(ajjUjj2)g=p1R10k1c(ar2)(r)p+1(1r2)np3 2dr 2dr=p1R10k1c(ar)(r)p 2dr 2dr=p1Beta(p+2 2;np1 2) Beta(p 2)Ek1c(aW)=(n1)1Ek1c(aW); whereWBeta((p+2)=2;(np1)=2).Thus,Hkisadiagonalmatrixwithentries(n1)1Ek1c(aW)initsmaindiagonal,thatisHk=(n1)1Ek1c(aW)Ip.This,alongwithEquation C{33 ,impliesthat,fork=1;2;3andl=0;1, Since(W1+ZZT)Wp(Ip;n1), Thus,returningtoEquation C{30 ,weseethat whichwhencombinedwithEquation C{29 gives Finally,weneedtocalculate 113

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2(XnXn)jj2)1oi: Usingtheresultsoftheorthogonaltransformation,shownintheEquations C{21 ,wewritetheaboveexpectationas 2(nXi=2YiYTi)1 2Ynjj2)1gi+a(11=n)1 2BEh(n1 2Y1)YTn(nXi=2YiYTi)1fc(jja1 2(nXi=2YiYTi)1 2Ynjj2)1gi; andduetotheindependenceoftheYi,i=1;:::;n,andn1 2E(Y1)=,thesecondofthetwotermsofthelastexpressionisequaltoamatrixofzeros.WenowcontinuewiththecalculationoftherstexpectationinEquation C{45 .WritingZi=(A+)1 2Yi,i=2;:::;n,yields (A+)1 2Eha(nXi=2ZiZTi)1IpZnZTn(nXi=2ZiZTi)1c(ajj(nXi=2ZiZTi)1 2Znjj2)1i(A+)1 2=(A+)1 2(aM2;0aM1;0M2;1+M1;1)(A+)1 2: RecallingtheresultsofEquations C{40 and C{41 andcombiningEquations C{45 and C{46 yields whichforthechoiceofa=np2isequaltoamatrixofzeros. 114

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4.2.1 withEquations C{25 C{43 and C{47 4.5.1 4{20 andusingthemultivariateversionofStein'sidentity,wewrite Now,usingthedierentiationproductrule,weobtainthat Wehaveprovidedanexpressionfor@[S1(XiXn)]=@XiinEquation C{19 andwenowobtainanexpressionfor@c=@Xi.Sincethefunctioncisgivenas 2(XiXn)jjc]+cI[jjk1S1 2(XiXn)jj>c] 2(XiXn)jj; itfollowsthat 2(XiXn)jj>c]; where 2=@Xi=c 2(XiXn)jj3@ @Xif(XiXn)TS1(XiXn)g: UsingthedierentiationproductruleandtheresultofEquation C{16 ,wecanshowthat @Xij(XiXn)S1(XiXn)T=2(11=n)fTj(XiXn)2fTj(XiXn)(XiXn)TS1(XiXn); 115

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@Xi(XiXn)S1(XiXn)T=2(11=n)S1(XiXn)2S1(XiXn)(XiXn)TS1(XiXn): Thus,bycombiningEquations C{51 C{52 and C{54 ,weobtain k1I[jjk1S1 2(XiXn)jj>c]1n1jjS1 2(XiXn)jj2 2(XiXn)jj3(XiXn)TS1: Further,wecombinetheEquations C{55 C{49 and C{19 toobtain 2(XiXn)jj2I[jjk1S1 2(XiXn)jj>c]: CombiningEquations C{56 with 4{20 and C{48 completestheproofofTheorem 4.5.1 116

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5.2.1 Now, ^BiEBi=~B(XiX)B(Xi)=(~BB)(XiX)B(X); anditfollowsthat Marginally,XiiidNp(;+A),i=1;:::;n,andthus WenowcalculateWiEf(~BB)(XiX)(XiX)T(~BB)Tg.NoticethatWi=Wn1Pni=1Wi=n1Ef(~BB)S(~BB)TgNow, and 117

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2ZiwhereZiiidNp(0;Ip).Wethuswrite 2Enn(k1)Xi=1ZiZTi1 2(A+)11 2n(k1)Xj=1ZjZTjo1 2=k21 2EnXi6=jZiZTi1 2(A+)11 2ZjZTjo1 2+k21 2Enn(k1)Xi=1ZiZTi1 2(A+)11 2ZiZTio1 2; anditisnoweasytoseethat 2EfZZT1 2(A+)11 2ZZTg1 2; whereZhasthestandardnormaldistribution.LetD1 2(A+)11 2.Nowtheexpectationinthelastlineof D{8 iswrittenas wheredijisthe(i;j)thelementofthematrixD,i;j=1;:::p,andziistheithelementofvectorZ,i=1;:::;p.ThekthdiagonalelementofmatrixQiscalculatedas whiletheexpressionofthe(k;l)th,k6=l,elementofmatrixQisobtainedas sincethematrixDissymmetric.CombiningEquations D{9 D{11 ,weobtainthat 118

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D{8 D{9 and D{12 followsthat Now,fromEquations D{5 D{6 and D{13 followsthat Further,wehavethefollowingtworesults FromEquations D{14 D{15 followsthat TheresultoftheTheoremfollowsfromcombiningEquations D{1 D{3 D{4 and D{16 5.3.1 InordertocalculateEf(EBnLEBc;n)(EBnLEBc;n)Tgwewrite EBnLEBc;n=~B(XnX)fc(jjk1S1 2(XnX)jj2)1g; 119

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2a1 2.NotethatmarginallyXiiidNp(;A+),i=1;:::;n.ConsidernowtheHelmertorthogonaltransformation 2(X1++Xn)=n1 2XY2=21 2(X1X2)...Yn=fn(n1)g1 2(X1++Xn1(n1)Xn)=(11=n)1 2(XnX): Then 2Yn: Also,YiiidNp(0;A+),i=2;:::;n.Recallthat~B=abk1VS1.Then, EBnLEBc;n=(11=n)1 2abk1VS1Ynfc(jja1 2S1 2Ynjj2)1g; andthus 2S1 2Ynjj2)1g2i: SinceVisindependentof(S;Yn),thelastexpectationcanbecalculatedas 2S1 2Ynjj2)1g2Vi: Inordertoevaluatetheinnerexpectationnoticethat 2(W1+ZZT)1Z: 120

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2Ynjjd=jj(W1+ZZT)1 2Zjj,whereW1Wp(Ip;n2)independentlyofZNp(0;Ip).Therefore 2S1 2Ynjj2)1g2=(A+)1 2Eh(W1+ZZT)1ZZT(W1+ZZT)1fc(jja1 2(W1+ZZT)1 2Zjj2)1g2i(A+)1 2: InEquation C{39 weshowedthat 2Zjj2)o=(n1)1Ek1c(aW)E(W1+ZZT)l1; whereWBeta((p+2)/2,(n-p-1)/2),k=1;2;3andl=0;1.Hence,combiningEquations D{22 D{23 D{25 and D{26 ,weobtainthat UsingtheresultaboutEfV(A+)1Vg,giveninEquation D{13 ,weobtain ThenalstepforcalculatingtheBayesriskistoprovideanexpressionforthecrossproduct Tothisend,wecalculate 2(XnX)jj2)1g: 121

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D{20 andwritingcforc(jjk1S1 2(XiX)jj2),Lnbecomes 2~BYn+(11=n)1 2BYn+B(n1 2Y1)on(11=n)1 2YnT(~B)T(c1)o: Now,n1 2E(Y1)=andY1isindependentof(Yn;S;V).Thus,Lnisequalto where First, Now,theinnerexpectationiswrittenas (A+)1 2Eh(nXi=2ZiZTi)1ZnZTn(nXi=2ZiZTi)1fc(jja(nXi=2ZiZTi)1 2Znjj2)1gi(A+)1 2 whereZiiidNp(0;Ip).Thus,combiningEquations D{34 and D{35 with D{26 weobtain 2(M2;0M1;0)(A+)1 2Vo=a2b2k2n1(np2)1Efc(aW)1gEV(A+)1V: ThisalongwithEquation D{13 showsthat 122

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2EhEZnZTn(nXi=2ZiZTi)1fc1g(A+)1 2Vi: UsingEquation D{26 ,weobtainthefollowingexpression 2E(M2;1M1;1)(A+)1 2V=ab(k1)Efc(aW)1gB: Hence,combiningEquations D{32 D{37 and D{39 ,weobtain Letan=np2andbn=n1(k1)1.Then,combiningEquations D{17 5{9 D{28 and D{40 ,andcollectingthetermsof,Bandtr(B)separately,Theorem 5.3.1 follows. 5.4.1 5{17 ,wewrite Considernowthefollowingexpectation 123

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4.2.2 ,theexpectationintherighthandsiteofEquation D{42 canbewrittenas Thecalculationofthederivativescanbeachievedbyusingtheproductruleasfollows Itiseasytoseethat Also,thefollowingequalityholdstrue whereXijisthejthelementofvectorXi.WewriteS=Pnm=1XmXTmnXXT,andusingtheproductruleagainweseethat FromEquations D{46 and D{47 followsthat, 124

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D{48 ,weseethat (@S1=@Xij)(XiX)=S1(XiX)(XiX)TfjtrfS1(XiX)(XiX)Tgfj: Itfollowsthat (@S1=@Xi)(XiX)=S1(XiX)(XiX)TS1trfS1(XiX)(XiX)TgS1: TheresultinEquation D{50 ,alongwithEquations D{44 and D{45 ,showsthat Also,usingsimilarreasoningasinEquations D{6 D{12 wecanshowthat TheresultoftheTheoremfollowsfromcombiningEquations D{41 D{43 and D{51 D{52 5.5.1 5{21 ,wewrite 125

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4.2.2 ,weobtainanexpressionfor WenowcontinuebycalculatingthematrixderivativethatappearsinthelastlineofEquation D{54 .First,thedierentiationproductruleshowsthat Inordertocalculate@c=@Xi,wewritethefunctioncas 2(XiX)jjc]+cI[jjk1S1 2(XiX)jj>c] 2(XiX)jj: Itfollowsthat 2(XiX)jj>c]; where 2=@Xi=c 2(XiX)jj3@ @Xif(XiX)TS1(XiX)g: Further,itcanbeshownthat @Xij(XiX)TS1(XiX)=2(11=n)fTj(XiX)2fTj(XiX)(XiX)TS1(XiX); andnowitiseasytoseethat @Xi(XiX)TS1(XiX)=2(11=n)S1(XiX)2S1(XiX)(XiX)TS1(XiX): 126

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D{57 D{58 and D{60 followsthat k1I[jjk1S1 2(XiX)jj>c]1n1jjS1 2(XiX)jj2 2(XiX)jj3(XiX)TS1: CombiningEquations D{55 D{61 and D{51 weobtainthat 2(XiX)jj2oI[jjk1S1 2(XiX)jj>c]: Further,similarcalculationsasinEquations D{6 D{12 showthat TheresultoftheTheoremfollowsfromEquations D{53 D{54 D{62 and D{63 127

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128

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GeorgiosPapageorgiouwasborninLarnaca,CyprusonJanuary29of1978.Heearnedabachelor'sdegreeinStatisticsfromtheAthensUniversityofEconomicsandBusinessinAthens,Greece,in2000.AfterreturningtoCyprusandworkingforoneyearforamarketresearchcompany,hedecidedtopursuegraduatestudies.HereceivedaMasterofStatisticsdegreefromthedepartmentofStatisticsattheUniversityofFloridaandaPh.D.inStatisticsinAugustof2007. 129