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A decision theoretic approach to estimation of variances and variance ratios

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Title:
A decision theoretic approach to estimation of variances and variance ratios
Creator:
Kundu, Sudeep, 1964-
Publication Date:
Language:
English
Physical Description:
vii, 91 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Bayes estimators ( jstor )
Confidence interval ( jstor )
Estimators ( jstor )
Estimators for the mean ( jstor )
Frequentism ( jstor )
Multilevel models ( jstor )
Statistical discrepancies ( jstor )
Statistical estimation ( jstor )
Statistics ( jstor )
Variance ( jstor )
Dissertations, Academic -- Statistics -- UF
Statistics thesis Ph. D
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 88-90)
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Sudeep Kundu.

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University of Florida
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University of Florida
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Copyright Sudeep Kundu. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text







A DECISION THEORETIC


APPROACH TO


ESTIMATION OF


VARIANCES AND


VARIANCE RATIOS


By


SUDEEP KUNDU
















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
JINTVEPSITT VP F PTA n





















To my parents and teachers












ACKNOWLEDGEMENTS


would


like


to thank


Prof.


Malay


Ghosh


for


his


wisdom


and


patience while


guiding me towards my degree.


Moreover, he is more than


ust my advisor, lie is


respected friend and shall always remain so


would also like to express my gratitude


to Profe and Pro


ssors Alan Agresti, Ramon Littell, P.k


f.


Sencer


Rao of the Department of Statistics


Yeralan of the Department of Industrial and Systems Engineering


for their guidance


and


support


while serving on


my Ph.D.


committee.


A note of


thanks goes


to Prof.


Ken Portier for agreeing to attend my Final Examination on


such


a short


notice.


Thanks


also


go


to all


the faculty members in


the


S


tatistics


Department for imparting their vast knowledge to


me during the s


ix


years


I


was a


graduate student here.


I


take this opportunity to specially thank Prof.


S


cheaffer,


who went to a lot of trouble in clearing all the obstacles to bring me to the university as a graduate student.


I feel very lucky to


have gained


very valuable experience in my three


years as


a statistical consultant in the Consulting


Unit of the department under IFAS, and


would like to thank everyone there for hearing with me over the


years.


In particular,


I would like t


a thank Prof.


Li ttell


ag


ain for supporting me and teaching me the art


of consulting.


I am also


very


grateful to Mr.


Steve Linda,


who made my tenure


as a


consultant


so enjoyable.


must mention the support of my t


eachers in college, specially Prof.


A.


M.


Gun


and Saibal


Chattopadhyay,


who inspired me to


come to


this


country for


graduate


a












my late high school and early college years and


who is partly responsible for what


am today.


Also, thanks go to my wife, Aparna,


who while going through a Ph.D


program


herself encouraged


and supported me to succeed in mine.


Last,


but not


the


least.


thanks must go to all my oid and new-found friends for just their friendship.












TABLE OF CONTENTS page

A~C~,KNOW\LED G EM EN T.. ................................................. . . ... ii


ABSTRACT


CHAPTER


1


INTRODUCTION


VI


Literature Review Overview of this M


anuiiscrirt . . ....... ...


1.3 Properties of Chi-squared distribution . ESTIMATION OF THE NORMAL VARIANCE


Introduction The HB Modc


r 1


Interval Etstimation at c Numerical Results ......


ESTIMATION OF THE REGRESSION VARIANCE


IH trods tiao r for.... ....... .......
HB Estimator for the Reuce Model. Interval Estimation ofcr2.......


*. . . . . . . . ..
*. . . . . . . . ..
. *. . . . . . ..


. . ...... .
. ..*..... .
...... .


3.5 Numerical Results .. .. . ....... .

ESTIMATION OF THE VARIANCE RATIO


Introduction


Development of the HB Estimators Risk Dominance over 60 .... ..


SUMMARY AND FUTURE RESEARCH


flTfl/~1T A lT3TT' A T' C'TrTrT/TT ,


1.1 1.2


2


2.1 2.2 2.3 2.4


3


11


3.1 3.2 3.3 3.4


11 12 21 31

38


4


4.1 4.2 4.3


5


38 39 48 55 58

63

63


...63


70 86 .88


BIBLIOGRAPHY.............................












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

A DECISION THEORETIC APPROACH TO ESTIMATION OF
VARIANCES AND VARIANCE RATIOS By

Sudeep Kundu


A ugust


199


2


Malay Ghosh


Major Department:


St


H'ierarchical Bayes


atistics


HB


) es


timators of the variance and the variance ratio in nor-


mal models are developed with the ob multiples of the sample variance or th


ective o


f dominating the corresponding best


e ratio of the two sample variances.


We also


develop highest posterior density


HPD


credible intervals


for the variances in some


cases.


First,


point


and


ntervai estimation of the error variance


,say


,is


considered


in the fixed-effect HB estimators of


S balanced one-way normal ANOVA model.


are proposed


Two general classes of


which dominate the best multiple estimator


So


being the error sum of


squares) under the entropy


loss.


The dominance of the


proposed


estimators over the


best multiple


estimator is proved.


We then find two


classes of HPD


credible intervals based on the two classes of priors for


using all


the cell means and the error sum of squares.


Such intervals have smaller length and


greater frequentist


coverage


probability than the usual confidence intervals based on


S only.


Al so


other


classes n


, - .


HPD credible intervals


are oroviaed


have


Chairman:


S


the


oa"


which













estimators is proved and expressions for the risk dominance are given for both models. Also, HPD credible intervals are provided for both models.

In the next chapter. a class of HB estimators is developed for the ratio of variances


from independent random samples from two normal distributions.


Using an extension


of the class of hierarchical priors for the one-sample case, two classes of HB estimators


are developed.


We prove the ri


sk dominance of these estimators and provide sufficient


conditions under which a sub


class of these estimators dominates the best multiple


estimator.


For one of these subclasses.


the expression


for the risk improvement


also provided.


Numerical calculations of the percentage risk improvement of the pro


posed


estimators over the best multiple estimators in all these situations indicate that the risk improvement can often be quite substantial.


Is


HB















CHAPTER 1

INTRODUCTION



1.1 Literature Review


Decision


theoretic estimation in normal linear models


has


been


an active


area


of statistical research for a long time.


Ever since the appearance of Stein


1956)


seminal paper on the inadmissibility of the normal mean in three or higher dimensions,


considerable amount of research effort has been spent on simultaneous estimation


of


the means or the regression coefficients in general linear models.


Usefu


reviews of


the topic have appeared in Brown


1986) and Ghos


h


1992a).


In contrast to the mean estimation problem, literature on decision the


oretic


timation of the variance or variance components in normal linear models is not


es

SO


rich.


It was known for a long time that neither the


MLE nor the UTMVUE


best multiple of the error sum of squares when estimating the normal variance,.


The best multiple of the error sum


of squares was also the best equivariant


timator under


a group


of location-scale


transformations.


However, in


yet


another


fundamental work of Stein


1964), it


was


shown that this best equivariant estimator


was inadmissible under any quadratic


loss,


LQ(a, a) =


a )(a - a


2)2


11.1


was


the


say


es-









>1


usual estimator by virtue of having lower risk under any


quaciratic


loss,.


they were


non-smooth and.


therefore,


were not admissible under any arbitrary


quadratic


(see e.g.


Sa


cks,


1963).


S


tein


1964)


conjectured


that,


in the one-way


analysis of


variance


ANOVA


situation.


within


a class of


estimators equivariant only under a


transformation of scale, a substantial improvement may be obtained b


solution


y a


Bayesian


when the ratio of the number of unknown means to the number of obser-


vations


is sufli


cienti


y


large.


This


conjecture has


been supported


in the numeri


findings of Ghosh


1992b


Brown


1968)


showed that the usual best multiple estimat


or of the normal variance


was inadmissible under


a wider class of loss functions.


This paper offered


valuable


insight into the statistical problems involving unknown 1


ocatio


n and scale parameters,


while not attempting to propose any actual estimators of the variance which improved


substantially on the usual estimator.


Brown


1968) also argued against


the use of


quadratic


ioss


for


estimating


any power of


the


scale parameter,


in particular


variance,


variance


one of his main reasons


s biased.


being that


Properties of some alternative


the best equivariant estimator of the


loss


functions and the corresponding


best equivariant estimators


were also


provided in


this


paper.


The


main result of


Brown's paper showed that there


exists a unique


loss


function for the estimation of


> 0), namely the entropy loss,


L (a,


a


,a



c2a


zog(


a
a2a)


-1,4


1.1.2


for which the best equivariant estimator under a group of location-scale transforma-


loss


cal


the


2a


a


, (


a









3


Brewster and Zidek (1974) obtained a hierarchical Bayes


HB) estimator of u2"


by extending the results of Stein


1964).


The resulting HB estimator was minimax


and admissible within the ci


Proskin


ass o


1985) proved the admis


f scale-equivariant estimators for the univariate sibility of Brewster-Zidek estimators within the


case. class


of all


estimators.


Strawderman (1974),


using a similar technique, developed a


class


of minimax estimators and made


an attempt to prove the risk dominance of an HB


estimator over the


best


multiple estimator of the variance in


the one-sample


case.


However, as noted in Ghosh


1992b), Strawderman 's result


was incorrect.


It is


well-known


that


in the one-way


analysis of


variance situation,


under the


group of location-scale transformations, the best equivariant estimator of


under


the loss


a2)a


d"(a- a2


) 2


1.1.3


is given by


where c, = 2-"F(t((n -1)k +


2a


)


)


/F(j((n- -1)k +4a


),


where S is


the error sum of squares.


Also, this estimator is the


constant risk minimax estimator


under the same loss.


In addition,


C0 is the best multiple of


So


under any


arbitrary quadrat


Lq(a,


a


2"


02c"


) (a - o2a)2


1.1.4


where Q(a 2a


)


> 0.


However, within the bigger class of estimators


equivariant oniy under scale trans-


formations,


caSa


is an inadmissible estimator of


a2 under every single


loss


given


La,


of


02a


ic


loss


a2"


ceS"


Lo


) = Q(












A subclass of these HB estimators dominate cae under any arbitrary quadratic loss


given in


1.1.4) in the sense


of having smaller freqiuentist risk.


The estimator pro


posed


earlier by


Brewster and


Zi


dek (1974


when


a


= .


is a member of this subclass


these estimators.


The numerical findings of Ghosh


when ax = 1) indic


ated that often


these risk improvements


could


be quite


substantial.


Ghosh


199


2


used


a direct


argument to prove the risk dominance of the RB estimators over the best multiple


estimator,


whereas


Kubokawa


1991


used


a definite


integral


technique to develop


a class of scale-equivariant estimators (not necessari


y


B


ayes


whi


ch dominated the


best multiple of the error sum of squares.


All of the articles mentioned above deal with the


point estimation of the variance


or the variance component.


Tate and


Klett (19


5


9


provided the expression for the


shortest


confidence


interval


of


a2


that


depended


only


on the


sample variance,


Cohen (1972) allowed the presence


to those given by Brown


196


8


of unknown means and used the estimators similar


) to construct confidence intervals with the same length


as those


given


by


Tate


and


Klett


1959)


and


higher


coverage


probability.


Using


the estimators


developed by Brewster and


Zidek (1974


),


Shorro


ck


1990) developed


a confidence interval that


improved


upon


the interval


obtained


by


Cohen


1972).


Maatta and


Casella


1987


discussed


the conditional


properties


of


the confidence


intervals developed


till


then.


Nagata


1989


developed


confidence


intervals


using


Stein-type testimators and provided numerical studies of the


improvements in both


length and coverage probability.


M aatta and


Caseila


1990


provided


a very


review of the confidence procedures for the estimation of the variance tram a normal


ot


S .


goo


d













Overview of this Manuscript


In this dissertation.


we develop hierarchical Bayes


estimators for the variance in


a variety of


different situations.


We also develop highest


posterior density


HPD


credible intervals for the variances in some


cases.


Berger


1985) contains a discussion


on HPD credible intervals.


The hierarchical Bayesian models that are considered here


can be regarded as extensions of the hierarchical Bayes ideas of Lindley and Smith


and are available for example in Ghosh


(


1992a) and Strawderman (1974).


Chapter


2,.


we consider the estimation of


oa"


wheree


is the error variance


in a fixed-effects


balanced one-way


ANOVA. Ghosh


1992b) developed a class of HB


estimators


which dominate the best multiple estimat


every single loss of the form given in


1.1.4).


or of the sample variance under


We consider here estimation of ua under


the entropy ioss


given in


.1.2).


Under this loss,


the best equivariant estimator of


given by


ts


UMVUE


,da"


where


(
)


1.2.1


Two classes of hierarch


cai B


ayes


priors are


considered in Section


2.2.


Similar priors


were used


b


v Ghosh


1992a)


and


Morris


(1983)


for th


e simultaneous estimation of


means.


Conditions are also provided under which a subclass of these estimators has


smaller risk than daS0


We use a modified version of a theorem by Kubokawa


1991


to prove the risk dominance of the HB estimators over the best multiple estimator.


A n expression for the risk improvement is obtained for each


of these two classes of HB


1.2


3


197


2


',


In


is


dc =


2~ -


1(j(n


n


- 1


)k)


-- 1


2cr)


oa"


F( j(


k +












coverage than the minimum length and shortest unbiased confidence intervals based


on the sample


variance. b


, oniy developed by


Tate and Klett (1959) and discussed in


Maatta and Caseila


1990).


Also,


we deve


op credible intervals using the


formulation


of Cohen (1972) and Shorrock


1982,


1990


Al though this latter class of HPD credible


intervals


have the same length


as the ones based on S


only, they have higher coverage


probability.


In Chapter


3


we d


evelop HIB estimators


of the disturbance variance in


two


nested


regression models.


Ge


if and


and


Dey


1988a)


developed Stein-type testimators


the disturbance variance, but their estimators


were not smooth.


Risk dominance of


the HB estimators over the best multiple estimator is proved for both


the full and


reduced regression models.


For the reduced model,


priors similar to the one used by Ghosh


the response function.


we use a class of hierarchical


,Lee and Litteil (1990) for the


Also, HPD credible procedures are provided for b


timation of oth the full


and the reduced models.


In Chapter 4. our objective is to develop a class of HB


estimators for the ratio of


variances from two independent random samples.


Loh (1986


provided some adap-


tive versions of the maximum-likelihood


(ML)


and restric


ed maximum-likelihood


REML) estimators of the variance ratio.


G elfand and Dey(


1988b)


obtained


a pair


of non-smooth estimators for the variance ratio


which dominated th


e best multiple


estimator under any arbitrary quadratic loss.


Using an extension of the class of hi-


erarchical


priors for th


e one-sample case,


we develop two classes of HB estimators


for the variance ratio.


We


prove the risk dominance of these estimators


by using


for


a


es















Before concluding this chapter. we provide in the next section certain properties of

chisqureddistributions which are used repeatedly in idigoptimal Bysceil intervals and in proving the frequentist properties.


1.3


Properties of Chi-squared distribution


In this section.


we provide a series of lemmas,


which exhibit some properties of


the chi-square random variable.


Lemma 1.3.1


Let x


< yand m




Then.


P(xi


Proof.


LHS


0


X/2


e


-(j+Z2)


1 -


'7


-S


in
2


-1


71
2


dz1 d22


x/2

~J0


C


-(z1 Z2)


rfl
- a
-I


I n


-1


Ti
2


dz1 dz2


j2


C


-(Li +Z2


z'2


in
2


4,)


IN


Ti
2


2


dz1


17


2v.


in
2


Ti
2


J /2


2


e


-(Li +z2


,n 'zI


in


in


Ti
2


dz1


d2_


.9 -


c)


y'


)


xn


>0


y/I
0


dz1 dz,


'


2


PC


in
2


YE


dz1


d
hA)


2


d22 --


n


P(xi,


5 y) - P( 2,


P(


y/2 JO












J/y/2
or/2


r/2
0


C


IN


-(z1 +2


m
2


n
2


z1z2 ) 2


r
I I
L 2 -m


-1


d dz


O .


Lemma 1.3.2


Let


ft


x)


denote the pdf of


xI,


v > 2


) and let F',(


x


Pix;


1.3.1


& x ).


Then


f',(x


)


x )


is


'lix.


Proof.


Integrating by parts,


1.3.2


Hence,


x) - 1


1.3.3)


Then, using Lemma 1.3.1, and the expression given in (1.3.3), it follows that the ratio


ft (x)/F',(x) is


4.in x.


Lemma 1.3.3


L


et


fv(x


) denote the pdf of a x


2


random variable and F.


Then x fv(x


Proof.


Leth(


x ) =x


f',(x


Fl/(


x ).


Then,


h(x)


1
F,}x)


xe


-4/


x


2J7


v/2-


2


)


x)


ri a; I


S


Fu,


2


fLJ(


x )


x)


/


F(


1
2


x)


/


x ).


)


x


Is


X.


P(


xi,


I
'I
-l


) =


x ) =


-2(


x) =


x ) =


Fe(


2"/


Fe(


Fe(


x ) -


fv(


[Pu-2(


Fe(


1 in


vfv+2(









A


Lemma 1.3.4 Let F,1(x) =P(xl


4-


- .


Then, for 0 < w1 i


< W2, F,(w2z)/|F.iwiz ) is


mf. z.


Proof.


Let g(z) = F,(w2z) Fe(wiz). Then, differentiating with respect to


z, we


hae


w2fv(w2z)


F,(wiz)


wif,(wiz)F,(w2z)


(F,(wiz))2


(w2z )f,(w~z)
Fj(w2z)


(wiz)fr(wiz)1


F,(w1z;


(1.3.5)


using Lemma 1.3.3.


Lemma 1.3.5 Let F( x:; A ) = P xt x ) and F,('x ) = P 2 < x ). T hen, for A > 0,


Pro oh.


For A > 0, one has


ft' (x)
f9(x)


Therefore, for x1


< x2,


P (xigA5x2) PQ4xl z)- P(x. x) P (xZ5 xi)


g'(z)


nz)_


0,


F,(x; A )


t


F( x )


In


X


in


X.












JX2
0


jX2 ftI


z)dydz


fy' (yfz)dydz


JZ fXI -f(~f( - fx2A(yifxejz)] dydz


.0


(1.3.6)















CHAPTER


ESTIMATION OF THE NORMAL


2.1


VARIANCE


Introduction


Consider a fixed-effects balanced one-


way analysis of


variance (ANOVA) model


with homoscedastic errors.


As mentioned in Section 1.2,


a decisi


on-theoretic approach


is taken in this chapter towards point and interval estimation of


0la


a


> 0


knownn,


where


Cr


2


denotes the error variance.


We denote


by


8,


k .


and


n the error sum


squares,


the number


of


cells and


the number of


observations


per


cell re


sp


ective


We consider point


estimation


under


the entropy


loss


given


in


(1.1.2).


It is well-


known that under the group of location-scale transformations,


the best


equivariant


estimator under the loss


acr


2a)


= aoa- a2


a)2


is


given


by


c0S"


where


- 1)k' 2a)


/


--1)k+4a)).


Also, this estimator is the constant risk


minimax


estimator of


a


2a


under the same


loss.


In addition,


C0


is


the best multi


p


leaof


under any arbitrary


quadratic loss of the form given in


1.1.4).


However, within the bigger class of estimator


s equivariant only under scale trans-


formations,


caSa


is an inadmissible estimator of


a under ever


y


single L


p


LOSS given


1 .1.4).


Under this


loss,.


the


best equivariant


estimator


of


a


0"


g


iven


UMVUE,


di


S"


where


-I


J


< 1- ______________________ $1


2


of


1(n


I(


So


C.


in


b


yV


its


1 1


ly.


2-"


P()((n













The dominance of such estimators over d&Sa is proved by appealing to a theorem


Kubokawa


1991Y.


In Section 2


.3,


we find the Highest P


osterior Density


HPD) credible intervals for


or r2


) under two classes


of hierarchical priors.


We then demonstrate that


such


intervals have nice frequentist properties as well in the sense that they have smaller


length


and


greater


coverage probability


(in


the


frequentist sense)


than


either


minimum length or the shortest


unbiased


confidence interval


s based only


on .


error s


urn of squares.


Also, two more classes of HPD credible intervals are considered,


one of which include


1990


).


s as


These latter inter


its members the ones considered earlier by Shorrock vais have the same length but higher coverage probe


1982,


)ability


than the usual intervals based only on


S


An interesting review of interval


estimation


given


in Maatta and Casella (1990).


ection 2.4, some numerical calculations are provided to indicate the


extent or


risk improvement of such estimators over


daS0


Also, some simulation studies are


performed to show that the proposed classes of HPD credible intervals have a higher coverage probability than the minimum length and shortest unbiased confidence intervals based on S only.


2.2


Thie HB Model


Consider the following balanced fixed effects one-way


ANOVA model


a


2I


0!


the


the


of


a2


is

S


In


a2"









'3


, 0 = t81,.. .,Ok)


We consider the estimation ot um iinaer


tne entropy


loss given in


1.1.2


We first


consider


a hierarchical Bayes model


which


is constructed


byV using iii-


dependent normal priors


with known


means,


say zeroes,


and


a common


unknown


variance at the first stage for the k cell means.


In the second stage,


dififuse priors


are assigned to the first stage pri


or variance


and the error variance, cr


The models


considered in Brewster and Zidek


1974) and Strawderman (1971, 1974> are members


of this


class.


S


uch priors are u


sed for


simultaneous estimation


of means


by Morris


1983


and G4hosh


199


2a


).


The hierarchi


cal model


is


g


iven in


I


II


b


elow,.


where


-2


R


Conditional


on9&


= 0


R


and A


Y


and


S are mutually inde-


pendently with Y


~N(0,


(nr


I-1


and


S ~r


j
a
(n-1)k'


n


;>2):


Conditional on B r and


are marginally


independently


di stribut


ed,.


where


fR(r


cxr


-2"


0


a

2)


while


A


haspdf


-~(4-b)


Writing


U


= ; A ,


it is


shown in Ghosh


1992b) that a


version of Jeifreys


prior


1)ased on


I) and


II) only is a member of the


above


class of priors.


From I)


- H II)


one has


conditional on Y


S


R = rand U


= u.


1, .. ,


(I)


II)


III)


R


and


A,


A


N


(0,


A


R


has


pdf


fA(


A


A


(72


+


i)


E )T


= A



4


A =


-la


Ar)-IIs);


& ~


= y,












where


t


= nlit


We say


that


Z


~Gamnmal a


, ) i t


z) cc exp( -cc


)z -


> 0


, A3


>N0


and I


is the usual indicator


function.


conditional on Y


= yand


S


=s, the pdf of U


is given by


fu(uly,


+ u-f" -+ >g,)u ,


2.2.1


For the loss


1


. .2), the Bayes estimator of


0-2a=


R- is


given by (E(fta


Now, from -


1i1), we have


E.(Ra


7.


y, .s)


P({(nk


-- a+ 2


+


2a


)8


(


P(t(nk - a + 2))
F(j(nk- a +2+ 2a))


P(jnk


s +ut)


'5\ -a


1


- a+ 2


+ uw)-"


Therefore, under the above loss, the hierarchical Bayes estimator of


>0g)


is given by


eff(


, S =E(Ra


Y


,S)


F (~nk


P jnk


w),


2


- a+ 2


)


+ 2a


(E{( + UW)-ya IY


)


, SI


say.


_}


(2.2.2)


We appeal


to the following version


of Theorem


2.


1


of Kubokawa


1991


which


provides sufficient


conditions under which e


HBa


(Y


,S)


dominates the best equivariant


fz(


14


(iii)


has


pdf


where in = t /s.


Y


0.2a


a


a


S
2


,(


)


S"


P a


Log


z), a


s ) ~ u


- (


- a+


,sa(


b("-6)O


, S))-1












ePa.b.a( w )


.O~a( w ),


where


exp is t"--.


sexpk -;s


exp - yy1-


" W


)s i("-1 k-1 (fi""5


exp- 2y


'dy)ds
-'dy )ds


Then HBfaY


, S)


dominates eo,0( S) under the


loss


#1.1.2),


efg ( Y


, S)


cr2a


- 2


, S)


eo a(S)


nfl


for all 6, o.


Note that


oo.w


Scan be simplified further


doa(w)


fo*


f* -


flL'


f '",


exp(-


1s


1


+x)


exp( --js(1 + x))


I


2;(-2)


i(k~2)


S


sink-1 ink+a~-


dxds


dxds


'Nt
IF( nk-.-2a)
2


04


fW


4 if-


27(1+


xitk2) (1 +


nh
x) - -2


dx


0_,2.


using (2.2.1) and (2.2.2)


(2.2.3)


We confine our attention to the subclass of HB estimators HBa(


, S) of


@20


The


next theorem provides conditions for a under which to,02W of Theorem 2.2.1.


satisfies the conditions


Theorem


2.2.2


Let


2


2. Then, $a,a,a(w ) satisfies conditions ( a) and (b ) of


Theorem


2.2.1.


Proot.


We first prove condition (b).


For


every


0


< a< k


+ 2,


using


(2.2.1).


b)


15


re lx


Jo*


i.e..


ea a(


5)


cr2


< 0


as


2 -a


dx


2,,aw


we


doaf


W


< a


)y i


E g ,2


< k +












FI((nk


2))


2a))


pi{ uw it4k-al( 1 ait(n-1Ik-2) O- 1 uw / 1+uw /


( 1+uw r


(1+ 1+w ( 1+4uw


P(tj(nk


f 7" zi(4-1


F(Cjnk


-- a + 2 + 24) f" z#(k)( 1


Putting


tin)


A-=


and


+11)


w
1+ w


F(j(nk


-a+


P(tj(nk


2


2))


+ 2a)) P ~Beta( k-a-~-2 (n1)k 20)
'if 2


3 ( k-a+2


x


k-a+2


( n-i )k)


(n-1)k+2a


- 1)k)


P[Beta(k-a+2, (n-1) fBeta( k-a-+2, (n-l +42a


C-


)


v 1


(2.2.4)


where Beta(m, p) stands for a beta-variable with parameters m and p.

Since do~a(w) = V2,2,a( w), one has


#a(k)
dkO,a( w)


P [Beta( j(k


P [Beta( j(k


- at+2


- 1)k)


), t((n -1)k +2a)




P |Beta
'


I
2


P (Beta( ik, c


1)kc)


4


(2.2.5)


1)kc+ 2o)


The following lemma, proved in Ghosh (1992b), shows that csa,a(W)~ satisfies (1b)


of Theorem 2.2.1 for 2


Sa < k


+ 2.


9-a


I~(


i&3


2+


x


2 -"


2- c


P Beta(k- 2


(n- 1)kA


K


Vt


K.


3


2~ a


+


i o


- a + 2


m




- z d((n-1)k-2)dz


- a + 2))


- z )it("~1)k-2+2a)dz


- a +


B (


F( j(n


f(j(n - 1)k + 2a) P


), j(n


(n








F-


verify


condition


ia) of Theorem 2.2.1,


note


that


= '1:


so. as w


-- cc.


U --


1. From


2.2.4) j


wehave


P(j(n


0a,a,,c C


- 1


+ 2a


= d4


2.2.6


To show that


d W


istiin w


that


is,


7 in v, define


a new pdf,


c zik-1)


- if " - )k 2) go1(


1


(2.2.7)


Then,


for


0


I
~LA9
fr(z)


!Itj


In


z.


tince


(expectation being taken here with respect to the pdf given in


2 .2.7)),


it is


that is


T


in w.


We now


consider the second hierarchical model.


In this


case,.


k


2.


The basic


difference between this model and the previous one is that, unlike the previous case.


the first stage prior mean is taken as Mv, and a diffuse prior is assigned to M


in the second stage.


as well


The model is described below.


Conditional on S = 0


Al


R


= rand


A =


A,


Y


and S are mutually


independently with Y


~~ N(O,


(nr)-


I4)and


S ~r


1 i


6'_


2


,JC, (n


(II)'


Conditional on Al


=m,


R =r and


A-=


A,


& ~N(m1,.


(III)'


Al


,R and A are marginally independent,


with Al


~ U~nz


f


orm


-so, oo),


fnir)


cr


0


a < nk 1) and JA(


A


) c


A


b(nt


A


(4-6) (b6


< k+1).


Again,.


writing


U


= iA


we have from


I)'


- (III)'


To


limn


f (


z)


,a-a


, i


EC


1


in


1w,


(I)'


)


w ) = 2-4


- Z)-"


~i"


-i


F({(n


- 1)k


Ar )"Is);












conditional on Y


= .s ana


U


= if,


P Gamma (


-(s at),


I
a nk


- a 1))


(iii)'


conditional on Y


=yand


S


the pdf oftU/is given by


fu(uly,5)~ '~- (+ uw) YtnktlItol)u),.


(2.2.8)


where


w * =n ,


and


I! = -i.


The HB estimator


2cr based


on vi)'


- iii '


is given oxy


*
C aba(~


, S)


P (j(nk


k - a


- a+1))


E((1


+ 1


+ 2


a))


+ U W*)-a lY


4*g (


),


say


(2.2.9)


Using arguments similar to those in Theorems 2.2.1 and


2.2.2,


we now have the


following theorem.


Theorem


2.2.3


Under the


ioss


0.1.2), e;aa(Y


, S)


dominates eo aS) for


every


t2,k + 1).


Now, to do some numerical calculations


,expressions for


the risk improvements of


efafi(Y


,3>


forO0


la

+ 2 and


e*


a,(Y1


, s)


for 0


* a


ck


- 1


over


eo,1(S) under


(ii>'


18


1


E( Ra|Y


.S)


a


S


(


)


S"


~1


, SI


aF6


= y, S


- g)2/s


F ()(n














- a+ 2


1 +


'V



ft)


1


.;*(k-a)


_ z )4((f-i ik--2>d.


1


-(2


1+


k


-a+2


i)k


2


(n -


i)k fJ


Z2 (k-a+2
Ai-a


'C


1


1


z(n1)k


S


(n -


-(1 - (V) 1)k


where &a( 1)


2


nk


V (2-


-a+


2


fov


zI'


) ~-l


(2.2.10)


1


---i )


dz


The next theorem provides an expression for the risk difference of eoi(S) and


, S)


under the loss (1.1.2).


Theorem 2.2.4 Consider the model under which Y


and


S are mutually independent


with Y


~ N(&,7n


-1 2k ) and


'- ~-)


Then for 0


k


+ 2,


E,,0 [L(eoP(S),


a) -


L~eff(Y,5), uj)]


= E((nk + 2L )AL,a]


(2.2.11)


where


AL-, E


((1


- V)j60(V)


(n


- 1)k


+


Zog(1


V))


L


nk+2L


(2.2.12)


.1 r


n


r~ir n~) \ I rr~ r - 1 I


19


nk


n~k


dz


2 Ji("-adz


(n-


% a <


- ro,(


- ((n-1)k)


effi(Y


x _1,s.












Then


3)


=-G


.( V)


and


eHB


, S


1)


wnere


w4,V)


i-V


V)-=


vi)


p-v -Uk


.For an estimator Gi@(V) of


at, one has


G&'(V)


C-


)1


Git


a2


V )


- Log


(G(KV
, '


Gi@(V) - Log G -


log 4'(V) - 13


(2.2.13)


If we write


r77=


2


e.2 -1


rk


92,


then,


under the reparametrization (9/


a


,1),


S and


are independent


ly


distribut


ed


with


~ xf,_2


S


and T


Introducing the


dummy variable


L ~


Poz


V


and


GC


are independent conditional on


L with


VjL ~Beta (4(k +


2L


),


1


(n -1)k)


and GIL ~


N


nk+2L.


Then,


L( eo,(


= E*0,1 EB


G(1


n


17)


- 1


+ Zog( 1


(V ))


=Ee/i,1


(nk +


2L


E


((1


- v


(n


7)


- 1)k


+


og(1


The next


theorem provides an


expression for the risk difference of


eoai(S) and


a
C aai(~


,S) under the loss


1.1.2).


Th eQremf


2.2.5


ForO0




k +1 and k


>2,


Eeg2 [L(eodSt ,


I'


;,


0*


S
C aai(~


I . , 11* g. (- a 1.4 V


5ta(


20


and


--1


T


E'i


S)


a2


) - L~e (


'2


L


+ 2L


L


4d


C


= Gu,(


Egg


-- 4,


)4,1


,S),


le


0 T*


.i* '


1


1


,


i(Y


coa(


Eeg


E,/,4


~ yi(r/).


sson(r/),


, S),


- V)4,


nk


) - L(


-













~Poisson t


Ca('8


2a


anid ViL*


~~ Beta


-i


4- 2 L* 1,


1{n -


)


1~


-a-+l


p.)


x


:2


1


- 7')


1


n-Ilk n-i )k


The proof is omitted because of its similarity to the proof of the previous theorem.


2.3


Interval Estimation of


oa


This section is devoted to the comparison of different confidence intervals for


a-


based on the hierarchical models given in Section 2.2.


Using S alone, the minimum


length confidence interval for a with confidence coeffi


cient 1


- a is given by


C1(S) - {o2 :


S


oa


c2S


},


(2.3.1


where c1


> 0


and


C2 (


ci )


are obtained from


)k-+-4 (*


=fen-1)k+4 (


1
--


and


t
I C'
1
C2


fenrt


x)dx =


-cx,


2. 3.2)


being the pdf of


achi


-sqaure variable with v degrees of freedom


Ta teanG


Klett,


1959;


NIaatta and Caseila,


1990


).


On the other hand, the shortest unbiased


confidence


interval for


with confidence coefficient


-cx is given by


C2(S) = {p2: d


S


a


d2S


},


2.3.3


where di


S0


and


d


are obtained from


L *


21


* I'
a


V


4k


ana


dx


x


- 512


-x)


y di









22


intervals have smaller length as weil as greater coverage prba iy in the frequentist


sense) than the intervals describ


ed int


Consider first the class of priors given


in


I


III)


Also, we confine attention to


intervals of the form


C3(W


, S) =


{


a -


: aiq(W)S


K


Il


C- 2<()S ,


(2.3.5


where a1


> 0


and


a2 (


>01 )


are


arbitrary constants,.


while 0




<. 1. Recall


that


w


= T /


S


and T


=7n


z


kY2


first


find


within


the


class


of


all


intervals of


the


frm


2.3.5


the


optimal


sbo(W) which leads


to


the HPD credible interval based on


the prior


given in


(I) -


(III).


S


pecifically, the following theorem is proved.


Theorem


2.3.


1


Consider the class of priors given in (I)


- ( III) with ( n -i


k +b-a


and b


< k +


2 .


Then


,within the class


of intervals


of the form


(2 23.5


,the optimal


$(w) is given by $o(w ),


where $o(w )


(1)k~b-a+4


is


a solution of


1
ofw 01Fkb+2


w
4b(w )a1


f(n-1 )k+b-a41o )2


w'
Fkb-+2 p tw) 2 ,


2 .3.6 )


where fv(x ) denotes the pdf of


xv,


and Fe(x )


Proof.


As


before


, write R= (


c2-,


Then from


I) - (III), the joint posterior pdf


of Rand U


= A /(n + A) given Y


= yand


S


=315i


f(r,uly,


s)


c expf-


1


r (s


+ ut)]rif(n-aI


j(k-b)


Now integrating with respect to a in


(2.3.


7). one


vets


2.3


.1


and


2


.3


.3


V.


We


> 0


=Px


K


2.3


7)


) - (













N ext we need to show that


f(riy


S i is


lo


g-concave.


Sinc


e expf


- 1 s


:s log-concave, it suffices to


show t hat P


tr


is log-conc


ave.


Thiis


mmediate consequence of Lemma 1.3


Since the p


osterior pdf of R given Y


= yand


S


= s is iog-conc


ave. using


2.3.8


the HPD credible


interval for Rs within the class of all intervals of


the form


2.3.5


is given by (


I
a200(w)


I
ait~o(w)


)


where 00(w


is obtained by maximizing


,10~w~
I'


exp(


-1


7~;(- k6aiiP x _


C


w z idz


2.3.9)


with respect to $(w).


This leads to the solution


1


ai4o~w)2


f(n-1)k+b-a a1)


1


a2#o


w


n-m 6~" a240(w)


2.3.10


w)


which is equivalent


to (


2.3.6) since


f,(x


) denotes


the pdf


of


2,.


This


proves


theorem.


Next


we improve on


confidence


ntervais


of


the


form


2.3.1


frequentist


sense


using


a subclass


of


HPD


credible


intervals


described


Denote


the interval given in


2.3.1


and


2


.3


.2


) as IML, and let 4e denote an interval


of the form




(2.3.11


where ci1


and c2 are determined from(


2.3.2


). Also,


we denote by


L(I) the length of


an interval I.


23


.2.


tact


is


an


aiw


w)


the


and


2.3.


2


in


a


in


2


.3.5


.2 :c p


{


w)s


K.


ired"-lik+6-ai-1


Xic-6+2


lw)


).


FA-a-2













for a2 where 0o(


w ) satisfies


I( n-i )k+4 (


I
- Fkat2
W )C1 I


''I
t


jko( w


f(n-1i)k+44 (po(w c2)


wo(w


)c2


)


2.3.13)


We now prove the following theorem.


Theorem


2.3.2


For an interval


given


in


S2.3.12 ) and


(2.3.


13


with 0




L( I0o


I'M


and Pq:(


cr2


P,:(


a


a


C IML ) for all


Proof.


We appeal


to Theorem


3.1 of Kubokawa


i1991


in


the special


case of


chi-squared distribution.


By definition L(40,


< L(IML


).


To prove that


'$o


has at


least as large a confidence coefficient as 'ML, following Kubokawa (1991), verify


we need to


is I in iv and


1';t
urn 0o(W 111-co


f(n-)k44 (


1
air$Mw)


{ '", L~aj(w


)


>fxn-luk+4 (


I
a200(w)


113
a200(w)


Firs twe prove


(i


). Making


W -. 00,


we get from


2.3.13


= Jtn-1)k+4 (n..,4(


w )


t2.3.14)


)C2


Since


Jft- i5k44(


1


fin_-1)k44(


1


)


and


f


is a chi-squared


pdf,


we must


have


m


$Q(W) =


N ext t


o show that t0(


w)


is i n


w


differentiate both sides of (


2.3.13


with respect


24


'C


2


a


1,


)


(ii)


) =


1,


fn-1k+4 (un


1.


O(


Fk-a,2


< L(


6 I.,


w ) <'


4o(w


F


Fu (









25


where


A EA w ) and BE2B(w )


fJ'n-1 k+4 F-,

1 1
- -ft'n,_lks4 Fk-a+2


1 1 w
-f(n-1)k+4 f-+

f 1 )


(2.3.16)


(2.3.17)


To show that B > 0, we nave from i2.3.13),


1 w
f( n-1 oc14Fk-+2


>1-)
w


w


fk-a 2( hoc)


t 0c2 Fk-a+2( )


w


fk-a+2(ttoci


t40ic1 Fk-a+2(


2.3.18)


since xfr(x)/F,(x) is 4. in x for


every iv


by Lemma 1.3.3, and 0


K


1
C2


,~>1


Li


Now.


using (2.3.16) and (2.3.13),


A = f( n-l )k+4 1)- i2


1 f/n-1)k+4 9
'oC2 (nl-i)k 4( 60


1 f'-)+ coci f(n-1Ik+4(j


(2.3.19)


since xf'(x)/f,(x) =


iv -


29


-xw s


in x for every iv, and 0


< I


It follows


from (2.3.15) - (2.3.19) that d'(w) , i.e. tco(w) is 5 in w.


0


0


B =


e













1


f ( r-1)k+4


F~at2


w
c~


)


'/


(


w ,


c1


Q


0


w


Fk-a-2


w


wT


C1 Qo


>0


putting


m = k


- a-i-


2


(aK<


k


+ 2),


n= k


- TV y - c100


and


it,
C20o


in Lemma 1.3.1.


This completes the pro


of


of Theorem


To improve on intervals given in


2.3


.3


and


2


3.4),


put


a,


= d,


(i


=1,2)


choose


b = a-


2


and a


+ 4in


2


.3.


6) to get


#0 from


= f(n-1)k+2 (


1
rGa2


Fk.~a4 (


w ,od


2.3.20


'I


Then following the line of the proof of Theorem


2.3.2, one can show that


L(Isu),


where Jsa denotes the shortest unbiased


confidence interval given


and (2.3.4),


and


P,:


r2


EEIt0


)


> ,2


a2


E i


su) for all


0*


2


,where


dto


is


defined in


.3.20).


For the class of priors defined


)'


in


- (III)'


first analogous


to


Theorem 2.3.1


define a class of confidence intervals of the form (


2.


3


.5


)where #0


n)


is a so


ution


f(n-1)k+b-a+4 (,~wa Fk-b+1


w2
q0(w~a1


f~n -1)k+b-a,4


1 wn
#o(w ja2 Fk4i#(w ua2


2.3.21


Such intervals are HPD within the class of intervals given in(


2


.3


.5


under the prior


- (III)'.


Now putting b = a -


2


and a


a, = c,


(i =


I


, 2),


2


.3


.21) reduces


C,


/f


w


N I N
I,. 1~I- ~ I I ~I -' I I - r. -. I I-t


1


26


2 .


3


.2.


f~n-1 2


a,


and


2


I40)


in


2.3


.3)


of


(I)'


to


U,


'Na C,
-, I


Fk-a,2


- Fu


c290


c290


< k


,
#041


< k + 3,












intervals


of


Cohen


1972)


and


Shorrock


1990


which


have


the


same


length,.


intervals have smaller length and greater coverage probability.


The intervais proposed


Goutis and Caseila


1991


are different from ours although they acn


ieve


the same


obiective as ours.


However,


it is also


possible to


develop


HPD


intervals


using


the formulation of


Cohen


1972


and


Shorrock


1982,


1990).


Under their formulation.


the minimum


length confidence interval for


a


2


with confidence coefficient 1


-ca is


given by


a1 + c1 S,


2.3.23)


where ai1


> 0


and c1


> 0


) are obtained from


f(n-)k+44 --)


= f(n-1,k+4 (a


1
-tci)


and


I ~1 'I


x)dx =


1


2.3.4)


f,(z


being the pdf of


x


,a chi-sqaure variable with v degrees of freedom


see Maatta


and Casella.


1990


On the other hand, the shortest unbiased confidence interval for


with confidence coefficient 1


- a is


given by


:.a2S


I2




(2.3.25)


where a2


> 0


and


c2 (


> 0


) are obtained from


fin -1)k+2 (-


-


=f(n-1)k+2 (1


C2


)


and


I.
C
ft
(~2 +C2)


x )dx =-


- a. (2.3.26)


We now find a class of HPD credible intervals for e2 under the hierarchi


cal priors


considered


in


I


III)


)'


and


- (


III)'.


A


subclass of such


HPD credible inter-


vais have greater


coverage


probabilit


yV


in


the frequentist sense)


than


the


intervals


2 -. ~. - -. 2 : or)


by


97


our


{


aiS


S 32


I2


C"(


a2


a-


S) =


5 (


--- a, (


- (


C4(


f(n_1,k(


S) = {


c2)S},


a2 +


f(,-15k(












where c (


> 0


is an arbitrary constant,


while 0


< oFW)


The optimal


W) within


the


class


of


all intervals of the


form


2.


3


.27)


which


leads to the HPD


credible interval based on the prior


g


iven


in


I


- (


III


is obtained


from the following theorem.


Theorem


2.


3.3


Consider the class


of priors given in


I )


- ( III


with (n


b-


a >O


and b <


k.


Then,


*within the class of intervals of the


form (2.3.


27


the optimal


is given by #o(w ),


where wo(


w ) is a solution of


1
A(n- 1)k+b-a+4 4ow


\ w Fk-b+2 ow


f( n-1 )k+b-a+4 1
#0(w


+


w
P Fkb 2 o(w)


+


2.3.28 )


C,


where fv(x ) denotes the pdf of


The proof of this theorem


xV,


and F.


x


K


x ).


is omitted due to its similarity to the proof of Theorem


2.3.1.


Next


we improve on


confidence


ntervals of


the form


2


.3


.2


3


)


and


2


.3.24)


a frequentist sense using a subclass of HPD


credible intervals described in


Denote


the interval given in


2


.3.23) and (


2


.3.24)


as 'ML, and let 4s denote an interval


of the form


S (#(w


+ c


(2.3.29)


where c1


is determined from


2.3.24).


For the


special case b = a


(


+ 2


) ,


we have, from


Theorem


2.3.3.


a class of


28


1.


w


in


2


.3.


2


8


)i.


2a


{


K


2


a


< k


-1 )k +


x ) = P(


: #(w)s


i )s},












where


w)I


satisfies


f(n -1i)k+4


1


fin- 1)k44


w


+c Fk-a+2 w
+ ci 4o(w )


(2.3.31


+ c1


We now prove the following theorem.


Theorem


2.3.4


For an interval Ib given in (


2.3.30


and


2.3.31


with 0


< s(W ) <


L( Istj =


L( IML) and


a


2


el4I0


a


U


()


'MILs) for all


Pro of.


We again appeal to


Theorem 3.1


of Kubokawa


1991


in the spec:


case


of chi-squared distribution.


By definition L(40


) =


L(IML


.To prove that Is has at


least as large


a confidence coefficient


as 'ML, following Kubokawa


1991),


we need to


verify


#o(w) is


I in wand


ii m 4o(w) = a1,


f(n-l)k+4 (


1


IL,
~(w)


)


)Fk ( U,)+


First we prove (i).


M aking w -+


f(n-1)k+4 1
li e-.


4o(w)


so, we get from


2 .3.31


1
=f~n-)k+4lim.-.(#o(w) + ci)


2.3.32)


Since


f (n-I )k+4


1


f(-I)k+I4


I, c,


), and


f is a chi-squared pdf,.


we must have


'ZO(W)


= a1.


To show that


w


is T


nil)


,cons:


der wi


and


w2


such that 0 <


wi


< w2.


Here.


and 4o(w2), respectively, maximize


p-si


fr


0s


29


a1,


(i


)


(ii)


)-


tim


dot


Fk -a+2


Pg


--


d>o(


) Fk (


2 f(n-i*+4


do(wi









30


To show that eo(w1


< oo(w2), use Lemma 1.3.4 and Lemma 3.1 of Shorrock


1990),


which is stated here.


Lemma


2.3.1


L


et


ft


x )


and g(


x


be two urnmodal densities


and let


I


maximize


f#+C


x )dx and t = 4t, maximize f'+C


g


x jdx.


Th en, if


f


x )


g


x )


is an increasing


function of


Next to prove


ii


)


use (


2.3.31) to get


fn-1)k


1


hi


f(n-1)k+4 't~


\c w
--)Fk\4o(w)


+C1)


fAn-1 )k+4 f(n-i)k+4 (


Fk-a


1
'o


1
4'0


)


U)


Fk


Fk-a+2(


w
--o


;t-


ww
#to 4o + c1


)


't o


w
-tCi


w
- Fk Q+~ Fk-a+2


#0


0,


putting m = k


-- a


+


2


(a

2


),


U)


k


w


and


in Lemma 1.3.1.


This completes the proof of Theorem


2.3.4.


Also,


we may note that the class of


intervals given in (


2


.3.30) includes as


ts members the ones considered by


Brewster


and Zidek (1974) and Shorrock (1990


for the special


case


a= =2.


To improve on


intervals given in


(


2.3.25


)


and


(


2


.3


.26),


choose


a< k +4 in


2.3.2


8


) to get


'to from


1


f(n-i )k 2


w
'to


-.f(?n-1)k+2
0+


-Fka,4 c2


Po



+


2 .3.33)


C2


Then following the line of the proof of Theorem 2.3.4, one can show that Pq2(


6


- 2


and


Oa -


C


6 = 0


x, #f


g)


-a+2(


+2(


, y =


\ /
F









31


of


Ph


f(n-1 )k~b-a,*4 (
(W1


1


f(n-1)k4-b-a,4 ow


m


-b-1 (


) ea 1


\


w)
pr(W )


2.3.34)


+ c1


intervals are HPD within the class of intervals given in(


2


.3.


27


under the prior


III)'.


Now putting


b = a


2


and a


+ 3,


2.3


.34) reduces to


1


tn-I )+2


Fk-a+3


w


1


-1-


The resulting interval 4s satin


sfies Pz(


E 4s)


Fat.3


-


> P


(


a2


e Isa


w


for all


Numerical Results


Tables 2.1 and 2.


2


provide the numerical computations of the risk improvement


of ef1f (Y


,5) over the best multiple estimator


S (n -


1)k of


a2


given by (2.2.11


2.12


for


k


= 5,


10


72


= 2


,3,


1


,2


3 along with several


sets of values of


z = 1,. . .


k.


Tables


2.3 and


2.4 provide the risk improvement of


e*


I (


y


, S)


over


the best multiple estimator


S


(n --


)k of


a2


given by (2.2.14) and (2.


2.


15


for the


same values of kt,


i,


a and 9,,


2 = 1,


*..., k.


The numerical computations were done


using the mathemat


ical software Mathematica (Wolfram, 1988).


It follows from these tables that risk improvement of


can often


be quite substantial


even


for


k'as s


mail as


3 .


eafi( For fix


, S)


over


ed kand


S


a.


1


k


the risk


improvement


seems to be decreasing in


n'.


It is our


conjecture that the subclass of


,-Y. *~ ~


S


uch


I)'


2.4


U -


and


2 .


90(o


< k


= f(n-i m.2


(n -












= 10,


7n.


= 2


in


Figures 2.1


and


2.2, respectively.


The three


tines represent


percent risk improvement for a =1,.


11 ustrate


the results for the HPD credible intervals for


C.-


random samples


were generated for a fixed-effects balanced one-way ANOVA model with k = 5, n =2,


and


two sets of values of Op.


The values of


c1 and c2 given


b)


y


2.3.1


were first obtained


with


confidence coefficient


1


-Qa


= 0.95.


Then


was obtained


using


the expression given


in


2.3.13


This


was repeated


for


2000


replications for each


combination of Ic, n, a, and 9.


Tabe


2


.5 gives the probability


of coverage and Table


2.6


gives the


average


#0o from


each


of these 2000 replications.


The value of


is the ratio of the length


of the


HPD


credible


interval Is


to the


length of the minimum length credible interval I'M.


Although the improvement in


the coverage probability is not


very large,


a fairly


large reduction in


the length is


obtained.


k


To


32


2,


3 .


and


the


= 1


and


2.


3.2


0


do
















Table 2.1. Percent Risk Improvement of effl(Y


.5')


over


S (n2 -


1)k for


= 0 = 1

=I


Vti


Vti


V i


6;-=2i - 1


V .2


n=2


1


a


-6.6888 10. 0212 14 .5122 0.7027


=2


0.0000 11. 6014 13.1113 0.5051


n=3


a


=3


5.1374 11.9838 11. 1324 0.3391


a


=1


-6.3054 8.8562 10.9909 0.0317


a=2


O .0000 9.6882 9. 1853 0.0190


a=3


4.3272 9.3339 7.1180 0.0104


Table 2.2. Percent Risk Improvement of ef'21(Y ,S) over


S/


(nt - 1)kc for


9, = 0


a,


= 1


N/i


n=2


a=2


n=3


a


=3


Ii


--5.4041

15.3777


0.0000 15. 37 15


4 .4901 14.8772


a


=1


-4.7753 11. 8702


a


i=3


0.0000 11. 3846


3.7204 10.52 13


33


k =


10


k = 5


6, 6, 6,


=2
















Table


2.3.


Percent Risk Improvement of eg(Y


, S)


over


S


(n - 1')k for


k= 5


O,

9,


93


= 0 = 1

=I


9- =


V

V

V


2i -


Vti


n2


a=-i


-v


2112


- 7.2112

4.0271 6. 1421


=2


a =2


0.0000 0.0000 '7. 5384 4.5709


a=-3


5. 0982 5 .0982 9. 2252 3.0861


n2=3


a


=1


-6.4158

-6.4158 3.4787

2.2548


a


=2


.00

0.0000


6.16


58


1.4616


a=3


3.9909

3.9909 7.0O88 0.8415


Table 2.4. Percent Risk Improvement of e~~1(Y


,S)


over


S


(n - i)k for


n=2


a=1


a


-5.5902

-5.5902


=2


a


0.0000 0.0000


=3


4. 5481 4 .548 1


n2


a


=1


-4.8062

-4.8062


=3


a=2


0.0000 0.0000


a=3


3 .6538


3.6538


34


10


= 0

= 1


V

V


2


642


k =




































15 10


5


0









-5


-10


,..


a-i a-2


b
'I- - - a-3

I
*
*
I I:
I
I,*
*
Ii 'I
I: I! 'S
* 1p
'S
U 'S

I:


I N- I..* t
I t
*
-S.
I
I
I '2 - - - -
I
I
I


I I I I I I I


0


10


20


30


40


50


11


35


4-'

0)
U 1~ a)


60


































15 10


5


0









-5


-10


I'
4
----*- a-i

a-2 I; I; 'I - - - a-3
I,
Ii
9'* ji
*
II .I I,
*
II
* I
I, I
I
9, I
* I
II I'
* I
I. I
U* 4
H 4'
N.,.

* N
N
* N
I
I

S - -
I
S
S


0


10


20


30


40


11


36


0)
U I.
Q)


50


60

















Table 2.5. Probability of coverage for 4a when k =.5, n = 2 and 1


- a = 0.95.


9,-=
I, =


1


V i 0.9500 k/i 0.9510


a=2


o.9500 0.9510


a=3


o.9500 0.9510


Table 2.6. Average value of do for I4 when k =-,n=2ad1-a= ..


9- 1


9- =


I


V


2


V i


2


a=2


O .8897 0.9596


0.9138 0.9702


a=3


O .9353 0. 9796


37


n=2















CHAPTER


3


ESTIMATION OF THE REGRESSION VARIANCE


3.1


Introduction


chapter,


we address


the problem


of


estimation


of


cra


a


-> (ct


where


is the error variance in


a linear


regression


model


and


introduce


two


classes


hierarchical Bayes


HB) estimators for the two nested regression models,


= X131


+ X232


3.1.1


Y


- X11


+


3.1.2)


E.


These HB estimators dominate the best multiple estimator of So


sum of squares),


S


being the error


S) = cQao where


2 -a


F (j~n


Ii(


I


(n


2a


- p+ 4a


under the relative squared error loss


L


a.
'a) au


a /


e2a


_-1


)


2


Although the above loss is considered in this chapter,.


.3.1.3


similar results can be obtained


In


this


rl


and


Y


of


)


)'7


eo..(


= (


+ e


- p +












the larger class of estimators equivariant oniy under a transformation of scaje, eqgak is no longer admissible.


class of hierarchical Bayes


priors is


considered for each of the two regression


mode


is.


The


class


considered


for


the


full mode


is similar


to the


class


of


priors


given in


(I)


HII)


of


Sec


tion


2


.2.


Geif and


and


Dey


1988a)


provided


S


tein-type


estimators of


dominating th


e best multiple estimator


S)


.The best multiple


estimator


can be


derived


as the


B


ayes


estimator


of


a2


under


the


reference prior


r(1. .


a


-2


2


as given in Berger and Bernardo


1991


).


For the reduced


mode


Ghos


i,


Lee


and


Littell


1990


introduced


a modified


version


of


the priors


considered here for the estimation of th


In Section 3.


2,


e response function.


we develop a class of HB estimators for a2" for the full model. It


is


shown that a


subclas


s of these HB estimators dominate the best multiple estimator.


In Section 3.3


,we do the same for the redu


ced model. In Section 3.4,


we provide two


classes of HPD credible intervals for


a


2


,which have shorter length and higher coverage


probability than the minimum length and shortest unbiased confidence intervals based on S only.


3.2


HB Estimator for the Full Model


Assuming XTX to be non-singular, let us denote the least squares estimator of /3 based on the full model as


-1


A


39


Si


- (


eog(


. .0,,












The hierarchical model is given in (I)


- iIII) below,.


wnere


(I) Conditional on B


= r and A


= '.


Y


is distributed


as Y


N(X3,


(II) Conditional on R = r and


(III) R and


A =A,


43 ~- N(O, ( Ar )-i(XTX X)-')


A are marginally independently distributed, with


fRIr)


and fA(


A


) O( A~Ib1 + @146


Writing U


= n,


we have


TTJ(U) cz ii


Then,


(i) conditional on ,


S


R =r and U


=11,


((:1 - u)3,


r-(1- a) (XilX)-1)


(ii) conditional on 3,


S =sand U


=11,


R Gamma (si+ ut), in - a+ 2)) where t =P(XTX)fl;


(iii) conditional on 43 and


fu(uly)


S =s the pdf of Uis given by oc u ( 1 W)-2 "~ EI(014tt).


(3.2.1)


Now, tinder the


loss


3


.1.3),


the Bayes estimator of R"


=-c


-2a


is given by


40


*1


R


r


-4


ex r


P ~


N


T


-in


= s,


= B, R


I,);








41


and


Ij(n-a+4o+


- a +2)


2)
-E


Under the loss


3.1.3), the hierarchical Bayes estimator of


is


given by


HE eab (P,


E(R-I\3, Si


8)


E(R-2a


a F4(n


P j(n


2a


+


- a + 4a +


2
2


E


E


1+ UW


1 + UW)-


4ka,c(W),


say,


(3.2.2)


where w = t/.s.


The following version


of Theorem 2.1 of Kubokawa


1991) provides sufficient


con-


ditions under which HRf


(j3


S)


dominates the best equivariant estimator eo,0(


caSe


of 02a under the loss


(3.1.3).


Theorem 3.2.1


Suppose


4k a,6,a( W 4a,b,a(W


is$T


in w, and hlm ,,0(w ) = ko,a(w), where


ko,a(w) =


fo*


SQ exp(
s2aexp(


1


1
--S
0
1
--S
2


("-p) (n-p)


-'(f;')
-1 (fW""


exp(
exp(


--j
--1


)y )y


2
2


-1


-1


dy)ds
dy)ds


Ten


ea ,


(j3,


S)


dominates eo( S) under the


01ss


3.1.3),


E(R2" 3


(S


s]


S)


S


(


) -a


Si


S"


2cr


( a)

i.e..


S)=


osa


ig,


a,


- a+


S) =


ca ;


+UT) * b


Fj(n












Remark.


The above inadmissibility result extends to any quadratic loss


L Q(S,


a"


2cr


)(a-cr


a)2


, Q(a-


Note that


w) can be simplified further


f" :


fW"


f* w "


exp(
exp(


1
-
2
1
-
2


1


+


s(1 +


x) x)


) x ) x


2


-1s 3(n+2a)


-i


(n+4cr)


-'dxds
-'dxds


n+2
I


fe w


I


a:


I
2


1+ 1+


4.
2


(n+2a)


dx


n+4a idx


#2,2,a


w)


(using (3.2.1


and (3.2.2)).


3.2.3)


We will again restrict our


attention


to the subclass edIaa(/3,


S),


as i


t


was done


in Section 2.2.


The next


theorem provides conditions on a under


which


#,,,a


satisfies the conditions of Theorem 3.2.1.


Theorem


3.2.2


Let


2


a < p+


2 .


Then, ta,a,a(W) satisfies conditions (a) and


b)of


Theorem 3.2.1.


Proof.


We first prove condition (b).


For every


2


a < p+


2,


using (3.2.1),


simplify #$2,,(W


given in (3.2.2)


2 -


P a,a,a(W


L' j(n- a +


t j(n -


a


2cr


+ 2


+ 4a +


) foi


U


2


az
2


(1 +uw


(1 +


-1.


uw)~


I.


(n-a+2a+2)
(n-a+4a+2)


a j(n -


F'


(fl-


a


+ 2a


+


a 4tv 4-


2


) f17


rW


/(1+w)


1 w)


2


(


--


1


- z)


i(n-p 2a-2)


du


(3.2.4)


42


> 0


as


#oaw


2_a


1( 1(


w)


as


we


2 -


dii dii


) = Q


$o.a(


z)z)-


) fi"


) foi a













(n- a +


2a +2) P


(n-- a+ 4a +


B


2)


p-a+2
2 2


Beta(


P IBeta(


n-p+2a
2
n-p+4x


4-2


~"2


I


�WI


)


2cr


- p +4


P


a) p


Beta


Beta


p-a+2
~2


n-p+2a n-p2 4*


1


)
)


(


-rw


3.2.5


1


Since 40a


wn)


we have from (3.2.5


B


4) a ,a ,a (


P


B


et


p-a+2
-2


et


n-+2 n-p+r4a


(3.2.6)


Now, using Lemma 2.1.1


a,a,a


(w) satisfies part


b


of Theorem


3.2.1


The arguments required to show that 4S4


,a,a (


w)


satisfies part


a) of Theorem


are omitted because of their similarity to the arguments at


Theorem


the end of the proof of


2.2.2.


Exact Risk When a =


We will now obtain an exact expression for the risk of effB1Q3


S)


forO0


under the loss


(3.1.3).


Note that for a = 1,


Ep2 [C


-4 (o1


3) -co


2 )


2]


=-2/


(n - p


+ 2)


2 -a


it,
-1-'
2


43


+ -


w I


w)


1


-4-


(


n


2 -a


1


( n


/
WI


1


K


C


WI


(


1


et


w)


P


P


B


B


~,+4 " 2+a-


E,
-'
B,


et


1


)


1


3.


.2.


1


1




a< +


3.2.


7)


wi(


w/ w/


+ w


= dua(


P (


w/


B (


- p +


w),


+ w


4














S


n2-a+4


1+


Vav


p-.- 2


A-


z:(1


S


n - a + 4


1 +


p - a n - P


+�2


+


2


2


+


fl- p


+t2


V B(PG 2 ) 1


ftv


zWT(1i


_ y {f-9t2 )du
-- 7 in-+ )


S


fl-p


(3.2.8)


- C1 - Qa,(V)), +2


where


pa( V) =


2


a + 4


V ( -+ (1


fov


(3.2.9)


z T (1


- z i -+2d


The following theorem provides an exact expression for the risk difference of eoj(S) and ef71(3, 5) under the loss (3.1.3).


Theorem 3.2.3 Consider the model under which Y


and


S are mutually independent


with Y


~N( Xj3, 91)In and


S ~ah n_.


Then forO0< a< p+


-(eo,i(S) - cT-)2



-4
(e~1(flS) - a2)2J


= E(n +


2L


)(n + 2L 2)ALt,


(3.2.10 )


where


AL,2


E


(2L +


(n


a


- 2)#a(V)(1


-- a +4)n


- p


~1 )
+2) nA


2


+42


+2L


(3.2.11)


Du..,. | .1.Q


1,_ .


44


1


Ey [-


2,


T ..


1 -v


L


- z )if"-P'du


- Da-


ar\


,


VIVQ\ --2 TitT


-z)i"-?'edu


li_. r\












Then es(,S)


n( V


) =


G'i#.(V) and eHij,3)


(3-V)(1iaV)


=CGka(1% where te.(V)


.For an estimator G&'(V) of o


_-p 2


.one has


F


E0,,2 tL( Gl(V ), 9j1))


(Gi(V)


Epk,,1 [(Gv'(V) - 1)21


(3.2.12)


If we write


17= (


f-1 PTXTXP then, under the reparametrization (//, 1),


and T


are independently distributed with S


~ xi_,


and


T


~ 4(77).


If the dummy


variable L


Poisson~q), then


Sand


GC


are independent conditional on


L with


VIL ~Beta (t p +


2


L ), t(n - p and G\L ~


2i g


Then,


Ep/,1 r(G#P(V) - 1)2


E?0,,1 [(Gk(V) - 1)2


(n +2L )(n +2L +2) E


[#t


2(V) L]


- 2(n + 2L )Ef'i(V) L] 1


(n+2L)(n+2L +2) E V--


1
n+t2L +2)


2


L


+


2


n+ 2L +


(3.2.13)


Then, we evaluate


{ V.(V ) -


1


2


+2L}


2


1


-- 'a(VV-


n+2+


- P1i


/,san - Ial


2


r


45


and


-1


S


L]


E


2)


2


L


}


2L


E4,2


2a2


= E


= E


= E


n+


ri. n:s . am


.












Differentiating #a(V) from (3.2.9) with respect to V


we have


-p


4'jV)


-a+


2


2V


722(1


n -a+4


-V)J


~JV)


2V


(p


-e ?jV)


-a


(n


+


2


-a


-(n-a+4)V q(V +4)(1 -V)


(3.2.15)


Substitution of (3.2.15) into the


r -H.s.


of (3.2.14) yields


-V/)


2


p


2


_-a+2


(n T


-(n -a


+4)V


- a+ 4)(1 - V)


altV)


2V
+ 1 (V
n -a +4


2


(1


- V


2


+ 2L)a


(3.2.16)


Integration by parts gives


E [V(1 - V'2 4'(V) L


v(1


=J


=I


1)2(1


-v) 2


#'(v)dv /B(2L


n -p


1


B (v2


2 (1


ThzP
2


- v)T


4 (=) 46a(V)]0


C-


n~J-j T. -, --


46


2V


E


1


(n 7


(n - p+2)(n +


f-P)


C'


ir(


N


n 4. 9 f,


- 24,( V) )2


- p +


- (1-v) - p'(v)do B 22L


-v)2v


v


-->r








47


Substitution of (3.2.17) into (3.2.16) yields


--V)4rJ


V


2


) 2


P- a+2


1


-(n-a


+4WV


4-


(n-a+ 4)


(n-p


2V


p+2L


2


1


-V


2('n


(n+


- V)cka


2


) 2


2


2(p


7n


a+2)


-a+4


2


(nU-


p


2


+2L)


p-a+2
n-a+4


V


n-a+4


-V)}


2


(n+


2


+2L)


(


(2L+a


n


-2


-p+2)(n


(1


-v


p)


-a+4


4a(V)


(n +


2


2


1


+ 2L)


n2


- v


2


Ar,..


3.2.18


The theorem follows


now from


3.2.13),


(3.2.14) and (3.2.18).


Remark. It is proved in Ghosh (1992b) that A0.2 = 0


A1,2


> O for all


1, 2,..., and


> 0 for all a E (2, p + 2),


while A0,.


<0, for allO0


K


a <2.


So,.


when a =2 and


0


there is no risk improvement over the best multiple estimator of the sample


variance.


C


1


(n


n-a +4


2) L)


2


+2


L


1


(


E


(n


+


1


(


E


L]


V)


n


(


C)


n-


(


2L


E


+


a


(n


2


- a + 4)


L


Ai,a


77=


)-


- V)


l =


-p+


-p+


+2)(n+


-p+


+2L


V)4


-p+2)


a+4


- 2)(1


)(n


- p +


- p +








48


3.3


HB Estimator for the Reduced Model


We consider the following class of hierarchical priors for the reduced model.


Conditional


on ,


R


= r and


A


A


Y


is distributed


as Y


N(X3,


r-I )


conditional on ft


'3 ~


N


/,


K


V1


0


A


X


TX )-1


I)


(III)'


v1


R


and


A are marginally


ndependent,


with vi1


~' Uniform( RP ),


fR(r)


oc r


-2"


and


faA)


xcA~


1.


b(1 +A


-4( 4-b)


Writing U


= A


C


and


we have the following theorem.


Theorem 3.3.1


Under the model given in (I)'


- ( III)'


conditional on Y


= y, R= rand U


| ( 1


46u,1


+ u


(I)'


II)'


and


A,


Ii


= X X


C'1 C21


C'2 C22,


(i)'


N'


I.


Cl'


-1-


C12


A =


= u,


vi,


r)-6












conditional on Y


=-y and U


=ut,


~Gamma (s+t),




- a+ 2))


where


-T [(


22.1/32-


(iii)'


conditional on Y


= y the pdi! of U


is given by


fu (ujy)


3.3.2)


where


Proof.


The joint pdf of Y


,13, v1, R and A is given by


f~y,x3, vi,


r, A)


oc r


exp -


r
2


fly -


X/3fl2


x(Ar)? exp


-


(/31


Cl1 (/3k


+/f C2232


+ 2


(/3k


AAb(l


+


A


(3.3.3)


Write


X-32


y -


T


(xpx (ii_


+ rb3-p


T


( 3.3.4 )




49


R


+ m ~t-t-+


u)


t/s.


I
-i


-v1)T


S


- v1)


ul(r2-6)O


- v1)T


C1202


X0


y -


C ({3 -0












=(vj -/3


- CE/ u02)


C11 (v1 -i#1


-c-1 C1-232 )


n22/3


-3TCn1CQiC2


-31


- C2Cup2)


Cii (v1 -#1


- C'2132)


+-4-a 2


(3.3.5)


Integrating with respect to vi' in (3.3.3), and using (3.3.4) and (3.3.5), one gets


f(y,f3, vi,, A)


, e


r
2


T


S


C(/3 -p3)) (Ar)2 e


-+ ([C22 1432)


I }4 6


S


C (a -- i) + #Cu#


)<-2ba + A)-I(4b)


(3.3.6)


Next observe that


XiY


and


Cn#1A


so that


I)


50


=(v"1


xr


-


xr


~a


C" C2'


C12 C22


$2


K


+0{C


r (Ar) exp -


+ (a -0)


XIY


= X(Y












Then, substituting (3.3.7) and


3.3.8)


in the exponent from i3.3.6),.


we nave


(A- 3)


T


C (i3-Q)


+A f A02 C 132


Cu (it1


- x31) + (i2


T
-/32)


C22


(-3


-/32)


- #1)


C12 (/32


-132) + C.2


- (1'


+CC02


+ -4C1 (32


xC11 {1t


+ C;, C1212) + C;,'C, (32


-/f32)


+2[f31


+ C1-001212)


+ Cj iC1 (32


-132)]


Cu 1232


T


-132)


if T


22 (/2


- /32) + C.2


Cii (i1t


-/32)


0C2., (#2


-132)


+ A/3'C2 u32


where


r7=


T


-


1


Cf


+ C,' 02


-T


51


T


(i1t


+42


(it1


r1


T


)


C


-02


- (#1


- (#1


- #2|


+ (#2



- 9) + (02












T


(71-At


Cn (y -A )


+(1+ A) (32


1b


+


A


)


C 22. (32


+


A


-T
4
1 A132


C212.'


(3.3.9)


Therefore.


rVt


c


xexp


Ar )f A -6b ( + 4


rS


T


1 2


)


T


C


22. (12


1
1+ A


+ 1:+A3C2.32


(3.3.10)


and, hence,


+ C1-iC2


I,


it


r


-1


I


C11


0


(3.3.11)


Using (3.3.8), one gets (i)' from (3.3.11). to 77 and 132, one has


N ext integrating (3.3.1O) with respect


f(y,r, A)


c


r


exp


-r


S


+ a


l A132


C21/32


A
i A


I


I


(1+


52


I -


)


f(y,r',s3, r, A )


C11 (q -1i)


)


#1


0


-iC


221


A


1


~ N


9 "


+ (9 -51)


+ A) #2


+(1









53


and.


f(y,r,\)


uj(Peb)( 1


+uw


- *(n-pj -a-2)


(3.31)


where w = t/s.


This gives us the theorem.


Therefore, under the loss (3.1.3), the HB estimator of a.2 is given by


e* .Y


E( RjY ) E( R2aIy)


-Pi -a+


- Pi - a


+ 4a


+2) EV[1


- -2)


EF[(i


+tUw


+~ UW)-


'Y|


2c'Y


*,s,( ,


say.


(3.3.15)


We will again restrict our attention to


the subclass


* f
C
aaa


y)


.Using arguments


similar to those in Theorems 3.2.1 and 3.


2


.2,


we now have the following theorem.


Theorem 3.3.2 Under the loss (3.1.3), e;aJY ) dominates e1(S) for


every


aC6


[2,p2 + 2).


Note that e*,1(Y) can be simplified further as


*
C aai(~)


S


n - p1


f I


4 f u


2(


1 + uw)-("-P1~4du


1 +uw)~


"~14* d


feV


S


z (1


n -p1 -a +4 fV


&fl
-r 2 (1


-7)ff(flPt2)dz
-


S


)


a-(n


S*


- a +


* Uk(n


- z)U"-Pidz













V" (1


fl-p


+2


;Iv


Pt(


-z) 2


S


fl-p


+


2


(1 - ( )),


(3.3.16)


where


2


4*(V) =-


n


The following theorem provides an


v;(P2-a+2


C


1


1


exact


expression for the risk of


e* i(


3 .3.17) Y) for


a E(


O, P2 + 2) under the loss (3.1.3).


Theorem


3.3.3


Consider the model under which Y


and


S are mutually independent


with Y


~~ V( Xf, ro ) and


S ~ o2


2 _ .


Then for 0


Sa


< P2 +


~(eo,i(S) - 0)


-ao


-4 (f i


- o2


= Et(n +2L*)(n +2L*


(3.3.18)


where


EF


+


nt


a


-a+4)(n


-p9+


-1V)
2) n


2


2


1


+2L*


n2


- V


-p


+


(3.3.19)


~Poisson (jb x~Cmitt and VjL*


~Beta({


{Cp


+ 2L* ,


The proof of this theorem


is omitted because of the similarity of its proof to that


Cry, r~ C'


.54


C)


dz


Eg2 o


2,


A. ,


,


2


-V)i("-?'edz


_ yli(n-pa


zT(


- z)it"~P*Udz


{{n -


- a + 4 fev


+ 2)A*


2L*


- 2)4*(V)(1


p)).












Interval Estimation Qt.


This section is devoted to the


comparison


of different confidence intervals for


based on the hierarchical mo


dels given in


Sec


tions 3.2 and


3.3.


Using


S aione. the


minimum length confidence interval for


CT -


with confidence coefficient 1


- a's


given


by


< c2S


},


3.4.1


where ci1


> 0


and


C2 (


I n-p 4 (*


> c1 )


= fn-p+4 (


are obtained from


12


and


I
C'
C2


fn-p(


x


- a,


3.4.2


1


f,(x


being the pdf of


XL,


a chi-sqaure variable with z' degrees of freedom


Tate and


Klett


1959:


Maatta and


Casella


, 990).


On the other hand,


the shortest


unbiased


confidence interval for J2 with confidence coefficient 1


C2(


&2.:di


& 25


- a is given by


},


3~.4.3)


where di


> 0


) and d2


) are obtained from


1


(


- -+2


1


)


and


1i


In_,(


x )dx =


- a.


We now find a class of HPD credible intervals for


a2 under the hierarchical priors


considered in


I) - (III) and (I


)'


- (III)'.


We shall show that a subclass of such HPD


credible intervals have smaller length as well as greater coverage probability (in the


<)A 1


C) .4 C)


4r nra * *nt-~ 4- a - 4- - - -. - I


.' c-I- I I ~T 1fl I I *%

3.4


DO


0l


C1(


{72


S


K.


a


a.


f T-+2


S) =


: ci


> di


)dx =


S) = {


3.4.4








56


where a1


> 0


)anti a2


are arbitrary constants,


while 0 -


;i W)


-1. Recaji


that1W


and T


=T - X


Within


the


class


of


all intervals of


the


form


3.4.5),


the


optima


WV)


which


leads to the HPD credible


interval based on the prior given in


I) - 1


I


Ix


is given by


the following theorem.


Theorem 3.4.1


Consider the class of priors given in


I


III ) with n -


p


+


b - a


and b <


2


.Then,.


within the


class


of intervals of the


0orm


(3.4.5


,the optimal


is given


by


0(w ).


where


w ) is a solution of


fn-p~b-a+4


1
F 40(w)a1


ri-b+2 4 ( 1


1


fn"~p+b--" 4


w
'4o(w ) a2),


)a2 )


3.'4.6


where


x ) denotes the pdf of


and Fe(x


)


P( xi


The proof is omitted because of


ts similarity to the proof of Theorem


2


.3.1.


Consider the


special


case


b = a


2


).


If we now set ai1


= ci


and a


from


Theorem


3


.4.1.


we have obtained a class of HPD credible intervals of the form


CE A2t/(,,)f}l


where cdo(w


satisfies


40( ci p-a+2


40wc


Jn-p+r4


F,_t2


)c2


w)ci )


3.4.8


I
i-% *.~ ~ . -~


n - . . * .-% ,1, I -~ -~ - . ~ -~ .... ..


> 0


C


a:).


'#0


= {


for


.2


Ia


2 =-


cif,


w


.2


fn-p+4


3,4.


7)


> ai )


= T /S


do(


- (


F,-b+2


xt,


fv(


p +


4( w


< p +


: c14o(


\ 4o(


0(w












To improve on confidence intervals given in t3.4.


3


and


(3


.4.4),


choose a = -


and a


Sp


+ 4in


3


.4.6) to get the following


class


of


HPD credible intervals


ItPO(W)S


2


5(w Is},


3.4.9)


where


is obtained from


1 viw
F_, \d 1


1


n-+2


d


w


3.4.10


\2


Following the


line of


proof


of Theorem


2


.3.2.


it


can be


shown


that


L(Isu), and


)l


E 1t0


)


> ,:


C


I


sU)


for at


where


I


"0


d


enotes


the


credible int


erval defined


in


3


.4.9


and


3.2


.14) and 1su denotes the


shortest unbiased


confidence interval given in (3.4.3


and


3.4.4


Next, we improve upon confidence intervals of the forms given in


3.4.1


) --


(


3.4.4


using the class of priors given in (I)'


- (III)'.


Using an argument anal


ogous


to that


given in


Theorem 3.4.1, for a


Kp2


+ 2,


a class


of HPD credible intervals of the form


{ a2 : c1#o(w)s


a2


K c2ck(w)s}


3.4.11)


for 02 where qdo(w) satisfies


w V


In-;'+4 1ow~i


w
#o(w


Fpr-a+2


3.4.12)


)c2


is obtained.


Following arguments similar to


those given in


Theorem


2.3


.2.


tis class


of HPD credible intervals have shorter length and higher probabilit


y of covera


ge


than


the rnBimiirn length rnni 1~nc~ ini-srunl W~cari nn


nnlr aiIrpn in


241


'3


I


A; .-' '-4 ,,


-
Di


{r2


for


C2


2


Inp2


LC


40o


HPD


).


d2e


yod2


P,:(


1
f,_,44 F,
do(w )c2












where


wn) satisfies


1


fn-p+2


w


F.


ekoc1


1


= .,,


q$0d2


-, a4


w


3.4.14


ch dominate the shortest unbiased confidence interval based on S only given in


3.4.3


and


3 .4.4).


Numerical Results


now


provide some numerical


calculations


of


the percent


risk improvement


Eg


and


e


am 1


over


S(z


p


+ 2).


Figures


3.


1


and


3.2


give the percent


improvements


for n


= 10 and p


3, and for


15 and p =


7, respectively, plotted


against '77


2


a.2>' /TX TXI3 using the expression


g


iven in Theorem


3.2.3.


Figures


3.3 and 3.4 give the percent risk improvements for the latter combination with p1


5,'


and Pi


given in Theorem 3.3


=*5 and p2 = 2, respectively, plotted against '7 using the expression


.3.


When a = 2, our HB estimator is the Brewster-Zidek (1974)


estimator in the special case of only one cell mean.


for


cr -


06


whi


3


.5


We


of


and


risk


P2 =


=-2


oo(


dod2


= (































I -'
$
p I
4
I
t
p
S
a -- - ~- a=1
I
a=2
S
a -- -* $
:1 4
XI
p S
4
* I
I I
4 S I
p I
4
* I
p 4
I
4 5
I
* N.
S
I
*
* I
I
*
S
I
N.
S
- - - - - - - - - - . t
I
* -


' I I I I I I


10


20


30


40


50


11


.59


6



















2


Q)
U
IS

a.


0


-2









-4









-6


0


60
































S
I I

-. a -a.
3' a-i
* I _____ a-2
* I
I S
1~ ---3-3
I S
I. I
* 'I
I .I
*
'I
I a S
* I
I, I I
I
I I,,
I.
* I
Ii I
I
I I,
* 4
I; I
S
*
I S
*
*
* S
*
I
*
*
I N .
* t
* -4
I 4~
S .4.
U

I
a


10


20


30


40


11


6


0


10






8






6






4


-'p
C 0)
(a)
2.~ 0)


2


0






-2






-4






-6


0


50


60













61


---.--- 3.-


1


I I
S

I - - - as3
I I
'4

I
* 'I
I I
*
I,
* I
I, I I
I
I 'I
II *

I 4
* I
*
I 4'
*
I
I
'4
I '4 4.
'4
I
I
S
I - - - - - - - . - - - - - . - - a -
- - - - - - - -


40


50


60


a a'
I
I


6










4


2


-4.
C

) 3


0










-2










-4


0


10


20


30


'11







































.......* -1


1.5 1.0 0.5 0.0








-0.5








-1.0


'ft
I *
--- -
* ft a~ 'in-,
g
* ft
* S
I I
* ft
* ft
I
* ft
* ft
I I
* I
ft
I
I
ft ft
ft ft
I
4

ft
ft
4
I
* ft ft
ft.' ft
.5
S* - - - - .- -ft. - *. --a-.. -S.
a - - - - -


-1.5


6


2


2.0


4-'
C

C) 1.
Q)


0


10


20


30


40


50


11


60















CHAPTER 4

ESTIMATION OF THE VARIANCE RATIO



4.1 Introduction


This chapter


considers estimation of


0f


a2,


the ratio


of


two normal


variances


based


on independent


random


samples.


Under


any quadratic


loss,


there exists


unique best multiple of the ratio of the sample variances.


However, when the popula-


tion means are unknown, such an estimator has been found to be inadmissible under


any quadratic loss.


ndeed, Gelfand and Dey (1988b) have found


S


tein type testima-


tors which dominate the best multiple estimator mentioned above.


S


uch testimators,


being non-smooth , are themselves inadmissible.


In Section 4.


2,


two classes


of hierarchical Bayes estimators of


2f


2j


are developed.


The two classes of hierarchical priors are extensions of the ones


considered in Section


of this


dissertation and in Chosh (1992b) for the one-sample


case.


It is


shown in


Section 4


.3


estimator.


that certain subclasses of these HB estimators dominate the best multiple The risk dominance results are proved by two-sample extensions of certain


general results of Kubokawa


1991) in the one-sample case.


for the risk improvement is provided for one of the two


ciasse


Also, exact expressions ~s of HB estimators.


D eveiopment of the HB Estimators


a


2.2


4.2













statistic


is<(


Ii,


72,Si,


where YII


Zn",2Y, and Sk =


Zn't=1( Yk,


Consider estimation of


77=


cr42


2j.


6


then under the usual group of Iocation-


scale transformations, and the


a
- -1


2


(4.2.1


the best equvariant estimator of 77 is


2 - 5 ni + 1


SI 32


(4.2.2)


However.


t


is shown


in


Gelfand


and


De


y


1988b)


that


under


any


quadratic


Q(a2,


rj


a),


where Q(a42


,crj)


> 0


the


bt


equivariant


estimator


inadmissible estimator of ij, and


Is


dominated by both


= mmn So,


(n2


- 5)


E1


(ni +2)S2


= mmnSo


(n2


(n1 + 1


- 4


Zn2'


Certain generalizations of these estimators were also provided by these authors.


should


be noted


that


the estimators Sf


and


are analogous to the estimators of


Stein


(1964)


in


the one-sample


case


where


the


best


equivariant


estimator of


normal variance under the group of location-scale transformations was shown to be


inadmissible by a class of testimators.


As


discussed in


S


ection 4.1, such estimators


being non-smooth are themselves inadmissible.


We now develop two classes of HB estimators of


If


j.


Let


2f=


(k =


1, 2).


The first class of hierarchical Bayes priors is as follows:


64


loss


1, 2.


lOSS


sf


and


60


is


an


Y2


sJ


It


the


S2),


k =


a)=


So =


)Si


- fi)2


If na


L(y


L(a


Rf












Conditional on Ra= r> and A1 = ,M1


N (0


, (


A r


and


M2 ~


and Al2 are independent with


U


niform( -oo, cX)


(III)


Marginally, R1


A1 and R2 are independent with


1


fR1(rl


(ni +A)-


I


(4~-b1


1R2(r


2)


oCc


r


2-I2


0


a2 K 12


+ 1.


Writing U1


= A1


(n1 + A1


the following theorem


is now proved.


Theorem 4.2.1


Under the model given in


'I) - ( III), one has


conditional onY =g,


Sa= s,


Rk =rk(


1, 2) and U1


= u1, M1


and M2 are mutually independent with


- ui)gi, (niri)1


(


1


- U1


1)


conditional on


Sk = sk


1, 2) and U


= 1u1,


mutually independent with


II)


65


c


r


f A


A1


0


A


-tb~
I


2,


and


0


< 3


(i)


1


N


and


((1


)


(ii)


Y2,


and


R2are


1)~


k =


n2r2


?> =


k =


gu,


M1 ~


ai < ni +


N (


M2













(Hii) conditional on 1


= y1


and Si1


the pdf of U1 is given by


- (1-b.
fu(tniyi, s) ~'- it ( + uiw1) ~ktL4Ijf,1), where w1 = t1/s1.


(4.2.3)


Proof.


Based on (I)


- (III), the joint distribution is given by


Y2,1


2
ni~~2 fJ{rk
k=1


Air1 )1/2


-$m


C


2

k=1


-1a


) A~~bl

A1 )(4-bl)


Integrating (4.2.4) with respect to m1 and mn2,


one has


Y2,1


c


- ~~


1+


'2


2


(n1 + A4 )-


x exp


A1


-7-


_2
y


+


e


_a
4.


2
ti(


s



Ic-


)


(4.2.5)


Then, from (4.2.5),


we have


s1,ri, A1


oc ri-a'


A1


66


cc


e


2


e


'1
4


~4.2.4)


1-b.


= si,


X (


31


f (gi,


si, 32, mi, m2, ri, r2, )


f (gi,


31, 32, rir2, 4)


ni


n1 + A1


f (gi,












Putting ui1


1Ain (4.2.6),.


we have


'*ll -QI


I-b


u- 11


exp


2


{ 7211L 2p'1


-a
3-


(4.2.8)


Now, we have from (4.2.8),


I lU1g .si3 r 2


2


(4.2.9)


and, from (4.2.7),


) cce


-~'2Z


(4.2.10)


This completes the proof of the theorem.


Using the loss given in (4


?7a1 , ,=


.2.


1


the KB estimator of


E (R1 1
E(R~Y


si)s )


17=


E (R' '2


_


52 S2


R2 R1is

)


(4.2.11)


)


From (ii) of Theorem 4.2.1, one has


E (RI
E (Ry


2


2


52 32


)2
5


(n2


-- a,--


22 (4.2.12)


S2


Further, arguing


as before,


E (R1 ?1 E (R Y1


Si


Si )


+


4


E [(1 UJ1W1 E ((1 + UT11


-'Wi1


-2


1


Si


ft -- a1 + 4


fgUi(l-b)(l + u~v1s-nf"ia+)du


(4.2.13)


Combining (4.2.12) and


4.2.13), the KB estimator given in (4.2.11


of n simplifies to


s1. ri, u )


67


xcr1


c exp


f


SI SI


= M/(ni -


f (gi,


f (ri, u1 #1, 31)


(r2 32


.
E ( Ri2


ni - 21









68


The second class of hierarchical priors is as follows:


Conditional on M&. = m,


Re=7r>


er=


1,2


l and


S2 are mutually independent with


(nkrk)-


1)


and


~ r-1x _1,


(II)'


Conditional on Ra= r and


- , MA


and M'2 are independent with


~Urn f orm( -so, so)


and


N(O.


tir2


(III)'


Marginally, R1, R2 and A2 are independent with


21


fRi (r


1)


r


r


c


0


2


0


K


ai1


+ 1,


a2


fAA(2)


c


A


-4b,
2


(n2 + ) ~


Let U2 = n~A


Then


fu2


(u2


c tL


Based on the above class of hierarchical priors, the following theorem is now obtained.


Theorem


4.2.2


Under the model given in


I)' ,


- < IIIy'


one has


conditional on


4 = -,


S5=-k,


R


k-=r4


k =1.2) and U2 = 2, A1


and M2 are mutually independent


with


)'


Ii:


1,


I> ~


mk,


S


and


1,0.


~1


and


0


K b2<


'1


3 .


(1%'


A2 = 4,


k =


~ N


A2 =


< n1


< n2 + 2,


M2 ~


fR2(r2













conditional on Yb


(kc = 1.2) and U2


=Ilk,


= U,


are mutually independent with


Gamma si,


1
2~i


- ai


+ 1))


Gamma (C2 + itt2) ,5(n2 - a2 + 2)) ,


t2 = n2


(iii )'


conditional on ?2= Y2 and


S2 =- 2,


the pdf ofLU2


is given by


fu(u21&2, s) ~ t


I
2
2


)(1 + w2 n -2+ )g y ),


(4.2.15)


where


w2= t 2/s2.


The proof is omitted because of its similarity to the proof of the previous theorem.

Now, from Theorem 4.2.2, the Bayes estimator of 'r under the loss (4.2.1) reduces on simplification tor


E (R1 Y1
E (RfY1


SI S2


si) si)


E (RI1 2
E (Ry2S


(n2 -a2 -2) E (i + U2 2 2, (n - a1 +3)


S2


S2


)


( n2 - a2


- 2)


(ni- ai ) 1


2


(1 + u2W2)-I(n2-a2)du2


0 -)(1+ u2W2


)~ n2 2- ) 2


(i!)'


69


and


R, and R2


Si
S2


= sa


ata2,62


fo u













Risk Dominance Qver


The


main


focus


of


this


section


is


to investigate conditions


under


which


some


estimators


developed in


the


previous


section


dominate


(n2 -


5)


(n1 + 1), the best multiple of S1


S2


y = o2


for estimating


2f


Two general


results are derived in


this


section.


The first of these results provides the basis for


determining a1, b1


and


a2 so


that the resulting ijai,bt,a2 has smaller risk than that of


So under squared error


and hence any quadratic


) loss.


The second result provides


a similar b


asis


for determining a1


a2 and b2 so that the resulting f/


ai ,Q2


hias smaller


risk than that of to under squared error loss.


In order to prove the first result,


S2


) 4'1(W1) for estimating 7.


consider


Denote by


a class of


IL(


estimators 61(


r) and F,(x;


Si,


52,


Wi


r) the pdf and the


distribution function respectively of a chi-square with v degrees of freedom and non-


centrality parameter


T.


Also,


to simplify notation,


write


0) and


x) and F,(


x) respectively.


The first theorem of this section is as follows.


Theorem 4.3.1


Suppose t2


>_6,


#k1(wi) is i urnm #1(wi


in w1, and


n,.


Th ent1


S1


, s2,


W1 ) dominates So under the loss (4.2.1),


2 )


4.3


70


of


the


HB


6s


(


Si


s2


(


Si/


0)


as


a)


(b)


i.e.,


) =


f,(


) =


F,(x;















i0(wi)


(say).


Proof.


Write T1


1 1 1
4' a
- (Ti.


Then,


E,a a (&1 r7


- i)


41(wX) - 1}xi fnii(xi)fn2-1(x2)fi(wix1; ri )dxi dx2dwi.

(4.3.1)


Ths,


lim J[Jck (wi) -112-(1f2-(2F(iz;idi


Jo 00'


Ow(1i) - 1}fnii(xi)fn2-1(x2)Fi(wixi; r)dxidx2


00(i -1xi fn 1(xi)fn21(x2)fi(wixzi;mr)dxidx2dw1




+ jj2 {1(w1)ff -1} 4'(wi)fff1(xi)fn2(2)F(wixi;i )dxidx2dw1



S(wi - x i fa 1(xi)fn2(2)f(wi z; r)d d2dwi




+ / 1(w)/f2{(wi)u -1} fa, 1(x1)fn2-1.(x2)Ft(wixi; r1)dxidx2dwi,


71


dw1









'72


J 2{ i}wiS - 9fa1(xi)fn2-1(x2)F1(wix1;rTi)dxidx2
Jo Jo 2222


4.3.34


Since %'i(wj) > 0, the right hand side of (4.3.3) is > 0 if and only if


t1 (i)


f 00
0 r1) { ft ax, i( )3x2} dx1


Je a _( i) (w1; )(


(n2 -


xy2fa,_(x2)dx2} dxi


3)-'


((n2- 2 - -5))- fJ x fna 1(x1)F1(wix1; ri)dxi


(n2 -


5) foox rfa, 1(i)F1wxi ; ri)dxt


(n12 --


for allr71 >O,


(4.3.4)


where the pdf of X1 is given by


hri(xi) xc xifa1,(x1)F1(wix1; Ti) .


(4.3.5)


Forr' >r',


hrii(xi)
h,,(Cx1)


F1(wix,; r{')
F1 (WI X; TV)


Sin wix1, i.e in x1.


fo fo'* g fa, _1(x1)fn2-1(22)F1(wi z1; r1)dzi dz2
,
fo* fo* gfa,-1(x1)fn2-1(x2)F1(wiz1; ri)dzidx2


Je zi fa, _1(x1)F1(wix1; ri )dzi


5)E, (V-1)














fc xifs11(x1
2 - ' r


Xjfn,1(X1


(nt2 - )
fT fA x f ,_1(x1


( F wjr fi(y)dy (fel"i fi~y)dy


f1(xizi)dzidx1
fidzizi )dz1idx1


x e


(n2 -5) fgfn


fo* C X e3


2

2


mul21

12


C


-2


e


(x1 z1


(xl


ZI)


1
2


-'dzidx1

-I dzidx1


1
--1 (fjo


l+zx2 dx,) dz1


(n2 - 5)


fo W


(112 - 5) --


z.


z1
4
1.


-1


-1


e- nH1 z +1dx1) dz1


(


_1


1+z
2-+


r~ +2
2


)~ 2(


21
2
21
2


+ 1)dz1

2)dzi


)2


0"1 + zi) 2


1
Zf
1~
2


-1


dz1


Ct1i(wi).


We now inv< ,rigate conditions under which HB

(4.2.1). Specifically, we have the following theorem.


(4.3.6)


dominates So under the Inss


Theorem 4.3.2 Suppose t2


6.


If a2


= 2


and


2




a1


< 3, then the HB


estimator #$ 2


Proof.


dominates to under the loss (4.2.1).


From (4.2.14), putting a2 =-


2,


Jo


'73


dx1
dx1


f Wi
0


(n2


(ni + 2)


dzi


f(


= b1


- 5) fe"(1 + zi


1 4(1-a11. . ... . 1,__ , ,, .














(122 - 5)


fWi /(1+WI)aa )-(; 1


S2 (iii - a1 + 4) foWi/(1 WI x#5>-''(1


1(722- 5) P (Beta (j(3 - a1), i~n 1)) 32 (nl + 1) P (Beta (j(3 - a1), B~ni + 3))


< -~
- lw)


4 .3.7)


Now, using (4.3.6), one has


x )Pftl~dui


S2 (11i + 1) f al/(ltol X2'-(1


Si (n2 - :3) P et \( 2,\vn 1) - 1+w 32 (n1 + 1) P (Beta () 1(ni + 3)) i- t


S1 (n2 - 5)P fBeta (*, j(ni + 1)) g i) S2 (nl + 1)p [Beta(', *ti + 3)) g


972,2,2


(4.3.8)


Then,


A
Pat ~72,2,2


P Beta (4(3 - a1), (nl + 1)) g P (Beta (4(3- al), iin + 3)) g


P[Beta (4, (n+ 1))
P (Beta (4, jrn1+ 3))


1


4.3.9)


SI


74


SI SW4


_ ,7-itni -n
- x)-i("I*Udui


5 g) 5 g)









- r


To verify condition (a), note that V1 =1 w/ (1 + w1 ),


so, as o1


-

- 1.


Then, for


a2 = 2


and a1 61


K 3, one has from (4.3.7),


im
wi -+ic


ep~w)=


n2 - r
ni + 1


(4.3.10)


To show that $1(wi) is T in wi, define a new pdf,


f,1(z)


Xc ze(1


-z) n"1-)ItoVI(z).


(4.3.11)


Then, for 0


vU


<-


vi, fv(z)/f,(jz)


OK Jtog4)(z)/Io~)


I


in z.


Note that


cE(1


,where expectation is taken with respect to the pdf given in


(4.3.11


).


Hence,


tti(wi


is { in v1 and, hence in w1~ (see Lemma 2(i), p.85


Le hmann,


1986).

Before deriving an expression for the risk difference of to and i5HB , note that 71a1a1, can be written as


Si (n2 -5) S2 (ni- ai+ 4)


Si (n2 -.5) fV1 xo1a-1)(1 S2 (ni1 - a1 + 4) f V1 XaQ(1


j0Wi/(l+Wi) xfz i )( 1


-x)-nI+1du


-x)4(fll-ldu,


S1 (
52 (li


- 5)


- a1 +4)


fl/I g(3-a1)(1 - zi"Uu


1 +


favi (10-ai(1


- 5)


*1+


3 - ai


2


n'i + 1 n2i + 1


1 (3-j)(1


-Vi)Pn(+1


fl/jX2 ai)( 1


Si 52


(n2


(ni


-a,1+4)


- Z)-41


#1(w1


-HB
Roman2


- x )i(n +n dui









76


Now, an expression for the risk difference of


'o0


and 77a,HB, is


obtained using the


following theorem.


Theorem 4.3.3


Consider the model under which and


S1i


are mutually independent


with ?,


1of) and S1


~ 2x_2


and Y2 and


52


are mutualLy independent


with }C2 ~- N(nm2, 72


-1 2
') C')
- a


1 and


S2 ~ a


22


where n2


>_


6.


Then for


2


<;


a1 < .3,


Em~y a


[csoy


2


-1


- 1)


- ('JIB


77


-1


-a)


nl + 2L1)(n1


(n2


- 5
-'3


)
)


AL


4.3.14


where


+1)


{i


2


2


+2L1


1 -
ni+


L


1


(4.3.15)


where L1 ~


Poisson (


1
I


nimt)


and V jL1


~Beta (j(i + 2L1


1~i-1


Proof.


Let T1 = n122 and GC


so that


+ T1/S1


Then60 =


G 1


S2 and


e


HB


- Gi19/'a


V1


52 where '.


V1


(n2-5(1-Vj)


n22-


5


1


- PI)(1 - 4,


V1


)


n1 + 1).


For an estimator G14'(V)


77, one has


Emst '


s2 -


,1,1 0( )


)U


21


4.3.16


=-E


21


AL


E


(2L1 + a1


n


V1


W1


).


Va1M)=


1


(


(


Ti


'a;


L (1(M


and


S2


S2


of


Em


S2


~1


21


77


--1


)


Em1/


ea,


(ni


) =


/


)


~ N(m1, n


+ 2L1


- B)


- 2)42,(B)(1


-ai+4


= T1 + Si


T1/S1


+W


i /su (G1@( B









77


Next observe that


Emni/a,i (1i( ) S2 - 1)2 = EEmi/atsu [(G19( ) S2 - 1)2 L1]



= E [(ni + 2L1) (ni 2L1 + 2) Emii/i,iu p2(v) )SQ L1}



- 2 (ni + 2L1) Ems /as1 {V( W) S2 L1} +1 =Ek(ni+ 2L1)(n1 +2L1+-42)xY


1 2
+ Ti' +2L1 2


4.3.17)


Then,


Em/01u .1


b.(1)
52


1 2 i 1( )
n1+2L1+2 2


1 2 n1+2L+2 L


Emn110,1, [(v'.2; 2 'jV)


Ei/aj ,11


017


(n2 - -) 1B 2 .( -# ()


(n2 - 5)(1 - I , '


- 2(4'.(V,) - &ai(Vi)) S2 Li]


/Si


Em














-2 Em~iaj


1


-1V I


71i+l


(


(14


1;/


E1
IC


1f,-O
n2-3


Em


,/a ,


2


'P


1


'(V1


n


-w


+1


1


It


+


1


-1
.1+21)


(4.3.18)


Now,


using arguments similar to those given in (3.2.15) - (3.2.18),


we have the


theorem.


It is shown in Ghosh


1992b


that A0,2 = 0, A,2


> 0 for all


S=


1, 2,... '4 ,


for ail a1


E t2, 3) and Ao,aj


< 0 for 0






The next


theorem is


aimed


at


proving


the risk


dominance of


6,jHB


over


for certain choices


of a1, a2 and b2.


The theorem is stated below.


Recall that


denotes the pdf of


x2


while fr(x; T) denotes the pdf of x2,(


7-


)a non-central chi-square


with v degrees of freedom and non-centrality parameter r.


Theorem 4.3.4


Suppose 72


;>6,


r42(u2


is{1


in w2, and


w-.x


Then 62($


S


1, s2,


W2 ) = S1qb2(W2 )


S2


dominates to under the loss ( 4.2.1 ),


E.22g


(


&


2


77


- 1


) 2


4
EmqZa2
1' 2


77


- 1


for all mn2,


2f


and o 2 it


78


)


S2


2


(p


1(


a1


, I)


6


0


fI


a:


(a)


i.e.,


2


) - di


24,


-6)


'b


(60














Pro of.


Write Tr, =


1 2 2
5n2rn2 0~-,.
A'


Then,


E~nm~a (62/


2
- i)


0 O {h~2)2 }2xifni-i1i n2-1(x2)f2(w22r2)dxid22.2


4.3.19)


Thus,


tim
f~7~co{
1v2 ~00


J
0


&
&1V2


Xi I~ 2(w2)2


2111n-(2F2Ex;2d iz


f1(x1)f1(x2)F2(w2xi-idxdx, FJo {~k2 112 nj-n2-2,2,1..


2 2-21}xnf--1(xi fn-i(x2)F2(2;T2)dxidx2d2 SjA ~OJO 2 2 2ii -,2-1\2X2 22\$2/\f2l




jj{~2 (w2)22 1}ini n1(x2)f2(x2;r)2(ix2r)d2d2



(4.3.20)


cn tbat


79


dw2














/ j2{ #~2(2) -1 i


X9fni,_(xi)fn,_1(x2)F2(w2x2;172)dxidx2 22


4.3.21)


Since #'2(w2)


< 0, the right hand side of (4.3.21) is


> 0 if and only if


I)2(W2)


~fo 1fi(i)fn2-)(22(w22; r2)dxi dx2
x2
fJ tfa,1_(xi)f2-(x2)F2(w2x2; r2)dxi dx2


_o xfn2-1(x2)F2(w2x2;2){foxlfn-i(xl)dxl}dx2 f& x2fn2-1(x2)F2(w2x2; r2) {J7 x fa1(xi)dxi} dx2


(ni


- 1


fe 00f21x)2(~ 27)z


)


(n1 - 1)(ni + 1) fjox2n2i-( x2)F2(w2x2;r2)dx2


f0


x 'ffai(x2)F2(w2x2; r2)dx2


ni +t I x-2fn2(2)F2(w2x2;r2)dx2


1


ni + 1


E,(X2)


(say),


(4.3.22)


where the pdf of X2 is given by


hr2(x2) ocx 2/ -(2)2w 2;2).


(4.3.23)


For <'>


(x2)


c


F2(w2x2; rs')
F2(w2x2; <)


I


Ifl W2X2,


ieinxz2.


Therefore, E(X2) is T in T2 so that (4.3.22) holds for all r2 > 0 if and only if


1


.7>


80


1


rI


hrI,


h i
r2(2)










81


fXe f( fn2-i(x2) f2(x2z2)dz2dx2 fo* f xs2 f12-1(x2) f2(x2z2)dz2dx2


f f'2e


-t -


e


- (x2z2)[1i


dlz2dx2


ni + 1


7ofo 2 2Ce


n, -1
-'---1
2X22


C


2(x2


z2 )5Cdz2dx2


lu>2


1


21 +


f v2


1.
-4.
C2
1
Z2


-1
(oo


Vt, -4 -a
!22(1+22)x 2
C

22z
2


dx2) dz2

dx2) dz2


1


ni + 1


702


- 4
"-'2
1.
zi


-1


1+z,
2


- 4


PC


S2


dz2


-1 *m-4 4
+Z2 )VP(~2~z~ )dz2
2 2


n22- 4
ni + 1


fjus2(1 +

f7'(1 +


-2

z2)~2


We now find conditions under which HB2262


(4.3.24)

dominates So under the loss (4.2.1).


Theorem 4.3.5 If n2


> 6, a1


= 2


and 2


< 3, then the H B estimator~"


dominates So under the loss (4.2.1 ).


Proof.


From (4.2.16), putting ai1


772,a,,a2


S n2- a2 S2 (ni +


- 2)


1)


Iou


-22(1 + u2W2)- Yn2-a2)du2


2


Thu 2)1+VW)i~2a-)


(n2 -- a2


(ni+ 1)


-2) fJ uW (-2 (1 + uW)~2


+ F du2


1 (2 2)(1-a2)) (1 LW2),.AA- 12 dzd,


1


_1


-1


z2
z.


dz2 dz2


= 2


SI 52


n1 + 1


<; a2 = b2










82


31 (, -5) P [Beta (t(3 - a2), j(n2 - 3)) S2 (lii + 1) P Beta (4(3 - a2), 4(722 - 5))


K v
5 -2


4.3.25)


Now, using (4.3.24), one has


SWCA


52 (ii1 + 1) fo72/(l+w2) xt'(1


Si((no - 4) fjo2 zat 52 (ii + 1) fJo2 xA'-1


_ sr in -sd
xtflb~2-s)dx


3 (72 -5) P [Beta ( , 4(r2 - 3)) i 12 S2 (nI + 1) P (Beta ( , j(n2 - 5)) 2)


r72,2,2


4.3.26)


Then,


#72,2,2


P (Beta (t(3 - 12), j(n2 - 3)) K V2] P [Beta (ft(3 - a2), 4(n2 -5)) K 1)2]


S


P [Beta (4, 3dn2 - 3)) P 21
P (Beta (j, j(n2 - 5)) 1)2]


N


1


(4.3.27)


by appealing to Lemma 2.1.1 again. To prove that #24,a2 dominates to, it now suffices to verify conditions (a) and (b) of Theorem 4.3.4.


- x))(n2-s)dz
- x)i(n2-7)dz













To show that

in u'2, define a new pdf,


) oc :f1


1 -


4 )(,-


I(O,v2)(


z).


(4.3.29)


Then


for 0


v2


<


2


fvi(


z)


/


fV(


z)


cchIo


(


z)


Igo,v2)(


z)


I


in


z.


S iflce,


,where expectati


on i


s being taken with respect to the pdf given


in


4.3.29),


it is


j. in v2 and, hence in w2 (


see Lemma


2


(i)


,p


.85


,Lehmann, 1986).


We now provide some numerical calculations of the risk improvement of bat.bia2


over


60


for


certain


values


of nIl


and


?2.


For n


2=6,


Figures 4.1


and 4.


2


give the


percent risk improvement plotted against


i


2cr2


) -1 nim for ni


= 3 and n1


respectively.


Th~e figures show that the risk improvement for values of r1 close to zero


can be quite substantial even for such small values of ni1


and 712.


The


HPD


credible intervals


for i


could not


be


provided


as they were done in


Chapters 2 and


3,


since our method of proof,


whi


ch used the log-concavity pro


p~erty


of the chi-square distribution function, could not be carried through in this case as


distribution function of an F


distribution does not have this property.


Nagata (1989)


provides expressions for confidence intervals of 77 based on Stein-type testimators.


83


z-


E


-22


Z)


(w2


the


fe (


= 7


= (


) oc


























-..--a1=


1


a=2

- -- a1 -2.5


/
I
I
'I.
I! *~
I, '
p
*
9 S
* S
* N
I


S.'


- - - - - - - - - - - - e - -n - -


p


40


50


60


84


6


4





2





0





-2


4-'
C
U S.
C-


-4





-6


0


10


20


30


1



















1.5 1.0


0.5 0.0




-0.5


- - --- a =-1
a -2
- - - a = 2.5


I
I
t


N


- ---~ -- - -- -


-1.0




-1.5


85


C C.)
It-


0


10


20


30


40


I .


50


t
1


60















CHAPTER


SUMMARY


AND FUTURE RESEARCH


In the analysis of linear models, point and


interval estimation of the error variance


play an


important role.


For the fixed-effects balanced one-way


ANOVA model,


obtained in Chapter


2 two classes of hierarchical B


ayes


estimators of the variance,


subclasses of which were shown to do sum of squares tinder the entropy loss.


inate the bes The numerica


t multiple estimator of the error 1 calculations that were provided


in Chapter


2 showed that the risk improvement could be quite substantial at times.


Also, our numerical calculations indicate that every member of these two sub


classes


of hierarchi


cal Bayes estimators is admissible.


Sutch an admissibility study will be an


interesting theoretical topic


for future research.


In Chapter 3


,hierarchical Bayes models similar to those proposed in


Chapter


were considered for two nested regression models


to obtain estimators which again


dominated the best multiple estimator under any quadratic


loss.


In both


Chapters


2 and


3,


highest


posterior


density (HPD)


credible intervals were provided for


error


variance.


A class


of


these


HPD


ntervals


not only


had


shorter


length


also had higher probability of coverage than the usual minimum length and shortest unbiased confidence intervals. In Chapter 4, two-sample extensions of the hierarchical


ayes


models proposed in Chapter 2 were used to


obtain two classes of hierarchical


ayes


estimators


of the


variance ratio.


A subclass of these estimators was shown


5


we


2


B


the


B


but


to









87


the treatment variance to the error variance are of interest.


Klotz, Milton and Zacks


1969) developed Stein-tvype estimators which dominated the corresponding ANOVA


estimators of the variance components, but such estimators being non-smooth cannot


be Bayes with respect


to any prior tinder quadratic loss.


Portnoy (1971


proposed


certain hierarchical Bayes estimators but could not prove any analyt


such estimators over the usual estimators.


Cal dominance of


It may be possible to extend our theoretical


findings to the estimation of the variance components or the variance ratio in one-way random-effects ANOVA models.

A multivariate extension of our results to estimation of the generalized error vari-


ance


n multivariate one-way analysis of variance models is also of interest.


Shorrock


1976), Sinha


1976), and Sinha and Ghosh (1987) proved inadmissibility


of the best equivariant estimator of the generalized variance under a variety of losses. Development of estimators of the generalized variance in a multivariate setting us-


ing a hierarchical Bayes


model remains an


open


question, and is


worthy


of future


exploration.


and


Zi


dek













BIBLIOGRAPHY


Berger, J. 0.
Springer


(1985) Statistical Decision
-Verlag, New York.


Theory and Bayesian Analysts.


2nd edn.)


B erger, J. 0. and Bernardo, J. M.


method. on Baue


To appear in


sian


Statistics:


D


edic


1991


oft


ated to


On the development of the reference prior the Fourth Valencia International Meeting


of


Morris De Croot.


Brewster, J. F. and Zidek, J.


V.


1974).


Improving on equivariant estimators.


A nnma/s


Brown, L.


of


D.


St ati


1968).


stics, 2.


problems with in]


21


-38.


Inadmissibility of the usual estimators of scale parameters in known location and scale parameters. The Annals of Mathe-


matical Statistics, 39, 29-48.


Brown, L.


D


(1986 ).


Fundamentals of Statistical Exponential Families wnth Applzca-


tions in Sta


tistical Decision


Theory.


IMS Lecture Notes Monograph


Series,


Cohen, A. (
bution.


1972 Jon


).


Improved confidence intervals for the variance of a normal distri-


irna? of the American Statistical A association, 67.


382-387 .


Gelfand, A. E. and Dey, D. K.


1988a).


ance in a linear regression model.


Improved estimation of the disturbance vaniJournal of Econometrics, 39, 387-395.


G elfand


, A. E. and Dey, D. 1K.


nal of


Statistic


1988b>.


On the estimation of a variance ratio.


Jo ur-


Gho


sh, M. (1992a). Hierarchical and empirical Bayes multivariate estimation. rent Issues in Statistical Inference: Essays in Honor of D. Basu, Eds. M. ( and P. K. Pathak. IMS Lecture Notes Monograph Series 17, 151-177.


Cur-


Ghosh


Gho


sh, M. (1992b)j. Report No. 40


7


On some Bayesian solutions of the Neyman,Dept. of Statistics, University of Florida.


Scott problem.


Tech.


Ghosh, M., Lee, L. C. and Littell, R. C.
estimation of the response function.


(1990). Statistics


Empirical and hierarchi and Decisions, 8, 299-3


cal Bayes 30.


Goutis,


C. and Casella, G.


mal variance.


The


Anna


991). is of


Improved invariant confidence intervals for a nor-


Statistics


,19, 2015-2031.


T{1c~fn I


IT


Cl


.7


1


.4 C, '~C, \


C I r.rr4 C v-.~.I...- I I*L-.** I .-... * r


Proceedings


the AMemory


The


9.


a? Planning and Inference, 19, 121-131.










89


Lehmnann, E. L. (1986).


Te


sting Statis


tical Hypotheses


2nd edn.).


Wiley, New


York


Lindley, D.


V .


and Smith,.


with discussion).


A.


F.


M.


Journal of the


(1972). Royal St


Bayes estimates for the linear mode


atistical Society B,


34, 1-41.


Loh,


w.


the A


1986). merzca


Improved estimators for ratios of variance components.


n2


Statistic


at Association, 81, 699-702.


Maatta, J.


M. and Casella,


of the normal variance.


G. (1987). Conditional properties of interval estimators The Annals of Statistics, 15, 1372-1388.


Maa


tta, J. M. and Caseila, G. (1990). Developments ind estimation (with discussion). Statistical Science, 5, 90-


ecision-theoretic variance 120.


Morris, C.
Infere


N.


nice,.


C. F. J.


1983). Parametric empirical Bayes Data Analysis and Robustness. Eds.


Wu.


Academic Press, New


confidence intervals. Scientific G. E. P. Box, T. Leonard and


York, 25-50.


Nagata, Y. (1989).
ratio of two vari


Improvements of interval estimations for the variance and


ances.


Journal of the Japan Statistical


Society,


19, 151-161.


P or tnoy, S.
model.


(1971). Formal Bayes estimation with application to a random effects The Annals of Mathematical Statistics, 42, 1379-1402.


P roskin, H. M. (1985). An admissibility theorem with applications to the estimation
of the variance of the normal distribution. Ph.D. dissertation, Dept. of Statistics, Rutgers University.


Rao,


C.


R.


1971).


Minimum variance quadratic unbiased


estimation of


variance


component s.


Journal of Multivariate Analysis, 1,


445-456.


Sacks, J.(
Math


1963).


Generalized Bayes solutions in estimation problems.


ematical St


atistics,


34,


75 1-768.


Shorrock,.


G.


variance.


1982). A minimax generalized Bayes confidence interval for Ph.D. dissertation, Dept. of Statistics, Rutgers University.


a normal


Shor


rock, G. mats of &


(
St


1990).
atistics


Improved c


onfidence intervals for a normal variance.


,18, 972-980.


Shorrock


R.


variance.


W. and Zidek, J. V. ( The .Annals of Statis


1976).


tics,


4


An improved estimator of the generalized


629-6


38.


'~*1' 1 n 7'


I fl-tn'


'-N *


C I ,. I


'.- . .~ I, - '4 III i&~ I I ~ ~ ~ ~* *~L~.- - I--.I .... I


Jon rnai


of


tieI


The Annals of


The An-









90


Stein. C. (1956). Inadmissibility of the usual estimator of the mean of a muitivariate
normal distribution. Proceedings of the Third Berkeley Symposium on Afathematzcal Statistics and Probabilit y, 1, 197-206.


Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal
distribution with unknown mean. Annals of the Institute of Statistical Mathe-


matics,


16, 1


55- 160.


Strawderman,


w


normal mean.


E. (1971). Proper Bayes minimax estimators of the multivariate The Annals of Math ematical Statistics, 42, 385-388.


Strawderman,


w.


E.


1974).


Minimax estimation of powers of the


variance of a nor-


mal population under squared error loss.


The Annals of Statist


ics,


2, 190-198.


Tate,
ol


R. F. and Klett,.


G.


w.


a normal distribution.


1959). Optimal confidence intervals for the variance Journal of the American Statistical Association, 54,


674-


682.


Wolfram, S.


1988).


M at hematica.


Addison-Wesley, Reading, MA.


f













BIOGRAPHICAL SKETCH


Sudeep Kundu was born on March 19, 1964 in Calcutta, India.


After graduating


from high school


in


1982,


lie


oined


Presidency College,


Calcutta.


He received a


Bachelor of


S


of Calcutta.


cience degree in A pri


He then came to


, 986, with honors in statistics from the University


the University of Florida to pursue graduate studies in


the Department of Statistics.


He obtained his Master of Statistics degree in April,


1988, and expects to get his Ph.D. in August, 1992.


As a graduate


student,


hewas at


eaching assistant for three years and worked as


a consultant in the Consulting Unit


of the Department of Statistics for the last three


years.


He has


been


a member of the American Statistical Association since


1990.


Upon graduation, he will be working as a Biometrician in the Clinical Biostatistics department of Merck Research Laboratories in New Jersey.













I certify that I have read this study and that iin my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.


Ma


r
'2 K~L
Llay Chosh, $~hairriihn


(V


Professor of


S


tatistics


I certify that I have read this study and that in my opinion


t


conforms to accept-


able standards of scholarly presentation and is fully


adequate,


in scope and quality,


as a dissertation for the degree of Doctor of Philosophy.


/7'
U


Alan G. Agres


V


/
K


N


ti '


Professor of Statistics


I certify that I have read this study and that in my opinion it


able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.


4
A


If flit'


1~~


Ramion C. Littelt Professor of Statistics


4


I certify that I have read this study and that in my opinion it conforms to accept-


able standards of scholarly presentation and is


fully adequate,


in scope and quality,


~V 2 rbcoar+~4,nn iV-~r *b~ Annrno i-S flr~r4-ar ,~C DLJ-...L..


conforms to accept-













I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.


Sencer Yeralan Associate Professor of Industrial
and Systems Engineering


Statistics in


the College of Liberal Arts and


Sciences and


to the Graduate School


This dissertation was submitted to the Graduate Faculty of the Department of


of Philosophy. August 1992


Dean,


Graduate School




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PAGE 1

A DECISION THEORETIC APPROACH TO ESTIMATION OF VARIANCES AND VARIANCE RATIOS By SUDEEP KUNDU 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 1992

PAGE 2

To my parents and teachers

PAGE 3

ACKNOWLEDGEMENTS I would like to thank Prof. Malay Ghosh for his wisdom and patience while guiding me towards my degree. Moreover, he is more than just my advisor, he is a respected friend and shall always remain so. I would also like to express my gratitude to Professors Alan Agresti, Ramon Littell, P.V. Rao of the Department of Statistics and Prof. Sencer Yeralan of the Department of Industrial and Systems Engineering for their guidance and support while serving on my Ph.D. committee. A note of thanks goes to Prof. Ken Portier for agreeing to attend my Final Exammation on such a short notice. Thaxiks also go to all the faculty members in the Statistics Department for imparting their vast knowledge to me during the six years I was a graduate student here. I take this opportunity to specially thank Prof. Scheaffer, who went to a lot of trouble in clearing all the obstacles to bring me to the university as a graduate student. I feel very lucky to have gained very valuable experience in my three years as a statistical consultant in the Consulting Unit of the department under IFAS, and would like to theink everyone there for bearing with me over the years. In particular, I would like to thank Prof. Littell again for supporting me and teaching me the art of consulting. I am also very grateful to Mr. Steve Linda, who made my tenure as a consultant so enjoyable. I must mention the support of my teachers in college, specially Prof. A. M. Gun and Saibal Chattopadhyay, who inspired me to come to this country for graduate studies. Although I do not know in what corner of the world he is in his pursuit of idealism, I must not forget to thank Dr. Azizul Haque who served as a role model in iii

PAGE 4

my late high school and early college years and who is partly responsible for what I am today. Also, thanks go to my wife, Aparna. who while going through a Ph.D. program herself encouraged and supported me to succeed in mine. Last, but not the least, thanks must go to all my old and new-found friends for just their friendship. iv

PAGE 5

TABLE OF CONTENTS page ACKNOWLEDGEMENTS iii ABSTRACT vi CHAPTER 1 INTRODUCTION 1 1.1 Literature Review 1 1.2 Overview of this Manuscript 5 1.3 Properties of Chi-squared distribution 7 2 ESTIMATION OF THE NORMAL VARIANCE 11 2.1 Introduction 11 2.2 The HB Model 12 2.3 Interval Estimation of tr^ 21 2.4 Numerical Results 31 3 ESTIMATION OF THE REGRESSION VARIANCE 38 3.1 Introduction 38 3.2 HB Estimator for the Full Model 39 3.3 HB Estimator for the Reduced Model 48 3.4 Interval Estimation of cr^ 55 3.5 Numerical Results 58 4 ESTIMATION OF THE VARIANCE RATIO 63 4.1 Introduction 63 4.2 Development of the HB Estimators 63 4.3 Risk Dominance over So 70 5 SUMMARY AND FUTURE RESEARCH 86 BIBLIOGRAPHY 88 BIOGRAPHICAL SKETCH 91 V

PAGE 6

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 A DECISION THEORETIC APPROACH TO ESTIMATION OF VARIANCES AND VARIANCE RATIOS By Sudeep Kundu August 1992 Chairman: MeJay Ghosh Major Department: Statistics Hierarchical Bayes (HB) estimators of the variance and the variance ratio in normsd models are developed with the objective of dominating the corresponding best multiples of the sample variance or the ratio of the two sample variances. We also develop highest posterior density (HPD) credible intervals for the variances in some cues. First, point and interval estimation of the error variance, say a^, is considered in the fixed-effects balanced one-way normal ANOVA model. Two general classes of HB estimators of a^" are proposed which dominate the best multiple estimator 5** (5 being the error sum of squares) under the entropy loss. The dominance of the proposed estimators over the best multiple estimator is proved. We then find two classes of HPD credible intervals based on the two classes of priors for using all the cell means and the error sum of squares. Such intervals have smaller length and greater frequentist coverage probability than the usual confidence intervals based on 5 only. Also, other classes of HPD credible intervals are provided which have the same length but higher coverage probability than the usual intervals. In Chapter 3, HB estimators of the disturbance variance are developed in two nested regression models. Risk dominance of the HB estimators over the best multiple vi

PAGE 7

estimators is proved and expressions for the risk dominance are given for both models. Also, HPD credible intervals are provided for both models. In the next chapter, a class of HB estimators is developed for the ratio of variances from independent random samples from two normal distributions. Using an extension of the class of hierarchical priors for the one-sample case, two classes of HB estimators are developed. We prove the risk dominance of these estimators and provide sufficient conditions under which a subclass of these estimators dominates the best multiple estimator. For one of these subclasses, the expression for the risk improvement is also provided. Numerical calculations of the percentage risk improvement of the proposed HB estimators over the best multiple estimators in all these situations indicate that the risk improvement can often be quite substantial. vii

PAGE 8

CHAPTER 1 INTRODUCTION 1.1 Literature Review Decision theoretic estimation in normal linear models has been an active area of statistical research for a long time. Ever since the appearance of Stein's (1956) seminal paper on the inadmissibility of the normal mean in three or higher dimensions, considerable amount of research effort has been spent on simultaneous estimation of the means or the regression coefficients in general linear models. Useful reviews of the topic have appeared in Brown (1986) and Ghosh (1992a). In contrast to the meaji estimation problem, literature on decision theoretic estimation of the variance or variance components in normal linear models is not so rich. It was known for a long time that neither the MLE nor the UMVUE was the best multiple of the error sum of squares when estimating the normal variance, say (7^. The best multiple of the error sum of squares was also the best equivariant estimator under a group of locationscale transformations. However, in yet another fundamental work of Stein (1964), it was shown that this best equivariant estimator was inadmissible under any quadratic loss, LQ{a,a^) = Q{a'){a-a')\ (1.1.1) where Q{cr^) > 0. The estimators that were developed by Stein (1964), which dominated the usual best multiple estimator, were based upon the acceptance or rejection of the null hypothesis that the population mean was equal to a specified value. Although these estimators, often labeled as preliminary testimators, dominated the 1

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usual estimator by virtue of having lower risk under any quadratic loss, they were non-smooth and, therefore, were not admissible under any arbitrary quadratic loss (see e.g. Sacks, 1963). Stein (1964) conjectured that, in the one-way analysis of variance (ANOVA) situation, within a class of estimators equivariant only under a transformation of scale, a substantial improvement may be obtained by a Bayesian solution when the ratio of the number of unknown means to the number of observations is sufficiently large. This conjecture has been supported in the numerical findings of Ghosh (1992b). Brown ( 1968) showed that the usual best multiple estimator of the normal variance was inadmissible under a wider class of loss functions. This paper offered valuable insight into the statistical problems involving unknown location and scale pajameters, while not attempting to propose any actu2Ll estimators of the variance which improved substantially on the usual estimator. Brown (1968) also argued against the use of quadratic loss for estimating any power of the scale parameter, in particular the variance, one of his main reasons being that the best equivariant estimator of the variance is biased. Properties of some alternative loss functions and the corresponding best equivariant estimators were also provided in this paper. The main result of Brown's paper showed that there exists a unique loss function for the estimation of cr^**, (a > 0), namely the entropy loss, for which the best equivariant estimator under a group of location-scale transformations is always unbiased. The entropy loss can be interpreted as the Kullback-Leibler divergence measure between two normal distributions, one indexed by the parameter a, and the other by as the population variance. But, the unbiased best equivariant estimator of a^" is also inadmissible. (1.1.2)

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Brewster and Zidek (1974) obtained a hierarchical Bayes (HB) estimator of a^" by extending the results of Stein (1964). The resulting HB estimator was minimcix and admissible within the class of scale-equivariant estimators for the univariate case. Proskin (1985) proved the admissibility of Brewster-Zidek estimators within the cleiss of all estimators. Strawderman (1974), using a similar technique, developed a class of minimeix estimators and made an attempt to prove the risk dominance of an HB estimator over the best multiple estimator of the variance in the one-sample case. However, as noted in Ghosh (1992b), Strawderman's result was incorrect. It is well-known that in the one-way analysis of variance situation, under the group of location-scale transformations, the best equivariant estimator of under the loss L{a,c^'') = u-^^{a-a^''f (1.1.3) is given by c„5° where c« = 2-°V{\{{n-l)k^2a))/V{\{{n-\)k^-Aa)), where S is the error sum of squeires. Also, this estimator is the constant risk minimax estimator of (7^° under the same loss. In addition, is the best multiple of 8° under any arbitrary quadratic loss L<^(a,a2'') = g(a2«)(a-(72°)^ (1.1.4) where C?(cr2<») > 0. However, within the bigger class of estimators equivariant only under scale transformations, CaS" is an inadmissible estimator of (t^" under every single Lq loss given in (1.1.4). Indeed, Stein-type estimators dominating Ca5° for a general a (> 0) were produced by Gelfand and Dey (1988a), when the number of cells equals one. Ghosh (1992b) derived a class of hierarchical Bayes (HB) estimators of cr^'* (a > 0) under the loss (1.1.3). Such estimators, unlike the Stein-type estimators, are smooth.

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r4 A subclass of these HB estimators dominate CaS° under any arbitrary quadratic loss given in (1.1.4) in the sense of having smaller frequentist risk. The estimator proposed earlier by Brewster and Zidek (1974) when a = 1 is a member of this subclass of these estimators. The numerical findings of Ghosh (when a = 1 ) indicated that often these risk improvements could be quite substantial. Ghosh (1992b) used a direct argument to prove the risk dominance of the HB estimators over the best multiple estimator, whereas Kubokawa (1991) used a definite integral technique to develop a class of scale-equivariant estimators (not necessarily Bayes) which dominated the best multiple of the error sum of squares. All of the articles mentioned above deal with the point estimation of the variance or the variance component. Tate and Klett (1959) provided the expression for the shortest confidence interval of that depended only on the sample variance, S. Cohen (1972) allowed the presence of unknown means and used the estimators similar to those given by Brown (1968) to construct confidence intervals with the same length as those given by Tate and Klett (1959) and higher coverage probability. Using the estimators developed by Brewster and Zidek (1974), Shorrock (1990) developed a confidence interval that improved upon the interval obtained by Cohen (1972). Maatta and Casella (1987) discussed the conditional properties of the confidence intervals developed till then. Nagata (1989) developed confidence intervals using Stein-type testimators and provided numerical studies of the improvements in both length and coverage probability. Maatta and Casella (1990) provided a very good review of the confidence procedures for the estimation of the variance from a normal population.

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1.2 Overview of this Manuscript In this dissertation, we develop hierarchical Bayes estimators for the variance in a variety of different situations. We also develop highest posterior density (HPD) credible intervals for the variances in some cases. Berger (1985) contains a discussion on HPD credible intervals. The hierarchical Bayesian models that are considered here can be regarded as extensions of the hierarchical Bayes ideas of Lindley and Smith (1972), and are available for example in Ghosh (1992a) and Strawderman (1974). In Chapter 2, we consider the estimation of a^", where a' is the error variance in a fixed-effects balanced one-way ANOVA. Ghosh (1992b) developed a class of HB estimators which dominate the best multiple estimator of the sample variance under every single loss of the form given in (1.1.4). We consider here estimation of cr^" under the entropy loss given in (1.1.2). Under this loss, the best equivariant estimator of a^'' is given by its UMVUE, d„S° where r(l((„-l)»: + 2a))Two classes of hierarchical Bayes priors are considered in Section 2.2. Similar priors were used by Ghosh (1992a) and Morris (1983) for the simultaneous estimation of means. Conditions are also provided under which a subclass of these estimators has smaller risk thein daS". We use a modified version of a theorem by Kubokawa (1991 ) to prove the risk dominance of the HB estimators over the best multiple estimator. An expression for the risk improvement is obtained for each of these two classes of HB estimators when a = 1. We provide some numerical calculations of the percent risk improvement and show that often the risk improvement can be quite large. The last section of the chapter deals with the derivation of HPD credible intervals. We obtain expressions for credible intervals which are shorter and have higher probability of

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6 coverage than the minimum length and shortest unbiased confidence intervals based on the sample variance. 5, only developed by Tate and Klett (1959) and discussed in Maatta and Casella (1990). Also, we develop credible intervals using the formulation of Cohen(1972) and Shorrock (1982, 1990). Although this latter class of HPD credible intervals have the same length as the ones based on S only, they have higher coverage probability. In Chapter 3, we develop HB estimators of the disturbajice variance in two nested regression models. Gelfand and Dey (1988a) developed Steintype testimators for the disturbance variance, but their estimators were not smooth. Risk dominance of the HB estimators over the best multiple estimator is proved for both the full and reduced regression models. For the reduced model, we use a class of hierarchical priors similar to the one used by Ghosh, Lee and Littell (1990) for the estimation of the response function. Also, HPD credible procedures are provided for both the full and the reduced models. In Chapter 4, our objective is to develop a class of HB estimators for the ratio of variances from two independent random samples. Loh (1986) provided some adaptive versions of the maximum-likelihood (ML) and restricted maximum-likelihood (REML) estimators of the variance ratio. Gelfand and Dey (1988b) obtained a pair of non-smooth estimators for the variance ratio which dominated the best multiple estimator under any arbitrary quadratic loss. Using an extension of the class of hierarchical priors for the one-sample case, we develop two classes of HB estimators for the variance ratio. We prove the risk dominance of these estimators by usmg a two-sample version of the direct integral technique of Kubokawa (1991). We provide sufficient conditions under which these classes of estimators dominate the best multiple estimator. For one of these subclasses, the expression for the risk improvement is also provided.

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Before concluding this chapter, we provide in the next section certain properties of chi-squared distributions which are used repeatedly in finding optimal Bayes credible intervals and in proving the frequentist properties. 1.3 Properties of Chi-squared distribution In this section, we provide a series of lemmas, which exhibit some properties of the chi-square random variable. Lemma 1.3.1 Let x < y and m < n. Then, P{xio Proof. Jo Jo r(f)r(f) ' ' — — 1 -—1 ^ * •» * rx/2 ry/2 ~ 3 yl ' ' Jo Jo r(f)r(=) ' '

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8 Jx/2 Jo y/2 fx/2 g-(*l+*s) x/2 Jo T{r^)m) n — m n— m dz\dz2 > 0 (1.3.1) Lemma 1.3.2 Let /i,(x) denote the pdf of f > 2 j and let F^{x) = P(x^ < x] Then fv{x)^ Fu{x) is [ in x. Proof . Integrating by parts, F^{x) = F,_2(x) 2^(x) (1.3.2) Hence, Ux)/F.{x) = \ [F._2(x)/F.(x) 1] (1.3.3) Then, using Lemma 1.3.1, and the expression given in (1.3.3), it follows that the ratio fi,{x)/F„{x) is I in x. Lemma 1.3.3 Let /^(x) denote the pdf of a xl random variaWe and F^(x) = P{xl < x). Then xf^{x)^F^{x) is i in x. Proof . Let h{x) = x/,(x)/F,(x). Then, h{x) = 3.g-x/2ji-/2-l F,(x) [ 2''/2r(i//2) ^fi'+2ix) F.{x) V 2 1 F.^2{x) F^x) using (1.3.2). (1.3.4) Since, F^+2ix)/ F^{x) is t in x from Lemma 1.3.1, the result follows.

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Lemma 1.3.4 Let F^x) = P{xl < x). Then, for 0 < toi < wj, F^{w2z)/ F^w^z) is i in z. Proof . Let g{z) = F^{w2z)^ Fu{wiz). Then, differentiating with respect to 2, we have W^Ujw^z) _ Wif^{wi,z)Ft,{w2z) F^{w^z) {F,{wrz})^ F JW2Z) \:^iz) {w2Z)fy{W2Z) _ {wiZ)fUwiz) FUW2Z) F^{wiz} < 0, using Lemma 1.3.3. (1.3.5) Lemma 1.3.5 Let F,(i; A) = P (xl,x < x) and F^{x) = P (xl < x). Then, for X > 0, F.(x;A) Proof . For A > 0, one has F.{x) T in X. t in X. Therefore, for xi < X2, P (xl,, < X2) P {xl
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10 0 (1.3.6)

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CHAPTER 2 ESTIMATION OF THE NORMAL VARIANCE 2.1 Introduction Consider a fixed-effects balanced one-way analysis of variance (ANOVA) model with homoscedastic errors. As mentioned in Section 1.2, a decision-theoretic approach is taken in this chapter towards point and interval estimation of cr^'* (a > 0, known), where denotes the error variance. We denote by S, k and n the error sum of squares, the number of cells and the number of observations per cell respectively. We consider point estimation under the entropy loss given in (1.1.2). It is wellknown that under the group of location-scale transformations, the best equivariant estimator under the loss Lia,a^'') = c7"''°(a a^'^f is given by CaS" where = 2-"r(i((n-l)fc + 2a))/r(|((n-l)fc + 4a)). Also, this estimator is the constant risk minimax estimator of a^" under the same loss. In addition, c is the best multiple of S" under any arbitrary quadratic loss of the form given in (1.1.4). However, within the bigger class of estimators equivariant only under scale transformations, CaS°' is an inadmissible estimator of (7^°' under every single Lq loss given in (1.1.4). Under this loss, the best equivariant estimator of a^" is given by its UMVUE, d^S" where Ti\{n-l]k) r(l((n-l)^^ + 2a)) (2.1.1) We aim at finding two classes of HB estimators of a-°' under the loss (1.1.2) which dominates a-°. The derivation of such HB estimators is given in Section 2.2. 11

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12 The dominance of such estimators over d„S° is proved by appealing to a theorem of Kubokawa (1991). In Section 2.3, we find the Highest Posterior Density (HPD) credible intervals for (7^" ( or o-^ ) under two classes of hierarchical priors. We then demonstrate that such intervals have nice frequentist properties as well in the sense that they have smaller length and greater coverage probability (in the frequentist sense) than either the minimum length or the shortest unbiased confidence intervals based only on 5, the error sum of squares. Also, two more classes of HPD credible intervals are considered, one of which includes as its members the ones considered earlier by Shorrock (1982. 1990). These latter intervals have the same length but higher coverage probability thaji the usual intervals based only on S. An interesting review of interval estimation of (7^ is given in Maatta and Casella (1990). In Section 2.4, some numerical calculations are provided to indicate the extent of risk improvement of such estimators over daS". Also, some simulation studies are performed to show that the proposed classes of HPD credible intervals have a higher coverage probability than the minimum length and shortest unbiased confidence intervals based on S only. 2.2 The HB Model Consider the following balanced fixed effects one-way ANOVA model = ^. + e,j (j = 1, . . . , n; t = 1. . . . , k), where the 0, are unknown fixed effects, and the e,j are i.i.d. N(Q,
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13 (Fi, . . . , YkV, 9 = (6-1,.. . , 9k)^. We consider the estimation of cr^" under the entropy loss given in (1.1.2). We first consider a hierarchical Bayes model which is constructed by using independent normal priors with known means, say zeroes, and a common unknown variance at the first stage for the k cell means. In the second stage, diffuse priors are assigned to the first stage prior variance and the error variance, a^. The models considered in Brewster and Zidek (1974) and Strawderman (1971, 1974) are members of this class. Such priors are used for simultaneous estimation of means by Morris (1983) and Ghosh (1992a). The hierarchical model is given in (I) (III) below, where R = a-\ (I) Conditional on 0 = 0, R = r and A = A, y and 5 are mutually independently with Y ~ N{e, (nr)-^ Jfc) and S ~ r-'^x(n-i)k^ (" ^ 2); (II) Conditional on = r and A = A, 0 ~ iV(0, (Ar)-ilfc); (III) R and A are marginally independently distributed, where R has pdf /fl(r) a r-5» {0 < a < nk + 2) while A has pdf /a(A) oc A-J** (n + A)-5(*-^). Writing U = it is shown in Ghosh (1992b) that a version of Jeffreys' prior based on (I) and (II) only is a member of the above class of priors. From (I) (III) one has (i) conditional onY = y,S = s,R = r and U = u, 6>~iV((l-u)y, {rnr'h); (ii) conditional onY = y,S = s and U = u, R ~ Gamma [^(5 -f ut), -{nk a + 2)] ,

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14 where t = nSjLij/,• [ ^^Y ^ ~ Gammaia, /5) if it has pdf fz{z) oc exp(— az)z'^"^/(o,oo)(2), a > 0, ^ > 0. and / is the usual indicator function. ] (iii) conditional on Y" = y and 5 = 5, the pdf of U is given by Mu\y,s) ~ u5('=-''Hl + uti;)-i("''-+^»/(o.i)(u), (2.2.1) where w = t/s. For the loss (1.1.2), the Bayes estimator of a-"" = i?-"* is given by [E{R° \Y , S)]-\ Now, from (i) (iii), we have E.iR-\u^y,s) = ra(n^--+ 2 + 2a)) ' ^' ^ ^(i(n^'-a + 2)) V 2 / r(|(nA: g + 2 + 2a)) Z^' A(nfc-a + 2 + 2a)) r(l(n^^-a + 2)) Uj Therefore, under the above loss, the hierarchical Bayes estimator of a^" (a > 0) is given by E{R'-\Y,S) /5N° r(|(nfc-a + 2)) , , 2/ Y[\{nk-a^2 + 2oc)) = ^a,6,a(ty), say. (2.2.2) We appeal to the following version of Theorem 2.1 of Kubokawa (1991) which provides sufficient conditions under which e^^^(y . 5) dominates the best equivariant estimator eo,c,(S) = d„S° of a^" under the loss (1.1.2). Theorem 2.2.1 Suppose Caj (f>a,b.a{w) T in w, and lim (^a,bAw) = d^;

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F • 90a('W) = : 1 Then e^^„(y, 5) dominates eo^a{S) under the loss (1.1.2), i.e., r2a r2a + In r2a for all e, a^. 15 < 0 Note that <^o,a(if ) can be simplified further as /o~ Jo exp(-i5(l 55"''-i dxds /o~ r exp(-i3(l + x))x5(*-2)55"^+°-i dxds = 2r(f ) lo x5(fc-^)(i + x)-^a,a,a{'^) satisfies the conditions of Theorem 2.2.1. Tiieorem 2.2.2 Let2a,a,a{w) satisfies conditions (a) and (b) of Theorem 2.2.1. Proof We first prove condition (b). For every 0 < a < ^ + 2, using (2.2.1). we simplify (^a,a.a{'^) given in (2.2.2) as o-c n\{nk g + 2)) Ji'-^m + uti;)-|<»^-°^^)(fu r(i(nfc a + 2 + 2a)) /J u5('=-»)(i + uw)-^^''''-''^^+^''Uu

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16 = 2" T{i{nk-a + 2)) T{\{nk a + 2 + 2a)) = 2 ^ rlf_uu^Ni(fc-a)(_!_U((n-l)fc-2+2a) u^^^ T(i{nk-a + 2)) Jo"a(''-°Hl-z)'^^"-^"-''^^ T{\{nk -a + 2 + 2a)) ^^(''-'(l z)5(("-^>''-'+'°>(iz ( Putting z = 1 + uw and V = w 1 + iv = 2" r(i(nA: a + 2 + 2a)) p ^Beta{!^,^^^^^f^) < v] X = 2' r(i(n-l)fc) P [5eo.a{w) = (i>2,2,a{w), one has <^a.a.a(u^) •^O.al^'') P [5e/a(i(fc -a + 2), |(n 1)A;) < v Beta(]{k a + 2), ^((n l)k + 2a)) < v Beta(\k,\{n l)k) < v 2 P (2.2.4) (2.2.5) P [Betaip, i((n l)k + 2a)) < v The following lemma, proved in Ghosh (1992b), shows that
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17 To verify condition (a) of Theorem 2.2.1, note that v = so, as u; — oo, V ~* I. From (2.2.4), we have lim 0a,aAw) = 2 :=rTr , o ^ = ^« (2.2.6) To show that <^a,a,a(t^) is t in w, that is, t in v, define a new pdf, Mz) DC (1 2)^(("-i)'=-2)/(o.„,(z), (2.2.7) Then, for 0 < u < v', ^ « T in Since 2. The basic difference between this model and the previous one is that, unlike the previous case, the first stage prior mean is taken as M , and a diffuse prior is assigned to M as well in the second stage. The model is described below. (I)' Conditioned on 0 = 9, M = m, R = r and A = X,Y and 5 are mutually independently with Y ~ N{0, {nr)-'^Ik) and 5 ~ r-\x(„_i)fc, (n > 2); (II) ' Conditional on M = m, = r and A = A, 0 ~ N{ml, {Xr)-'^Ik); (III) ' M, R and A are marginally independent, with M ~ Uniform{ -oo,oo), fnir) oc r-i" (0 < a < nk+1) and /a(A) « A-5''(n+A)-5(*-*) (b < k + 1). Again, writing U = we have from (I)' (III)', (i)' conditional on Y = y, S = s, R = r and U = u, 0 iV ((1 u)y + uylk, {nr)-\{l u)!^ + uk-'Ull)) ;

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18 (ii) ' conditional onY = y,S = s and U = u, R ~ Gamma (^2^^ 2^^^^ ~ + 1)^ > (iii) ' conditional on Y" = y and 5 = 5, the pdf of U is given by Mu\y,s) ~ J^'-'-'Hl + uu;-)-5<"'»-»+i)/(o,i,(u), (2.2.8) where w* = nE*Lj(i/, — y)^/^ and y = fc~^SjLi!/,. The HB estimator of (7'°' ba^ed on (i)' (iii)' is given by 1 E{R''\Y,S) /5n« T(\{nk-a + l)) V2/ r i(nifc-a + l+2a)) ^ I ' 1 -1 = -5" <^:,fc.a(^)' say (2.2.9) Using arguments similar to those in Theorems 2.2.1 and 2.2.2, we now have the following theorem. Theorem 2.2.3 Under the loss (1.1.2), e^^ JY, S) dominates eo,<,(5) for every a € [2,A; + 1). Now, to do some numerical calculations, expressions for the risk improvements of for 0 < a < A: + 2 and el^ .,{Y,S) for 0 < a < A: 41 over eo,i(5) under the loss (1.1.2) for different values of n, k and a are required. Note that for a = 1. e^^i(r, 5) simphfies to ^HB S) = 1 nlink -a + 2) z^(fc-°)(l z )t(i-^)''-Vdz ^'"''^ ' 2r(i(nA;-a + 4) j^^ ,^,i^-<^)(l z)'.(r.-i)k)^^

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19 1 nk — a + 2 1 nk — a + 2 1 + k a + 2 (n l)k 2 25(fc-<'+2)(l _ ^)i(n-l)fc (n l)k z)^(("-i)'=)rf2 (n where 4>a{V) [1 <^a(V)] (2.2.10) nk-a + 2 jv^^(^^_^^i^^^ The next theorem provides an expression for the risk difference of eo,\{S) and «a,a,i{Y,S) under the loss (1.1.2). Theorem 2.2.4 Consider the model under which Y and S are mutually independent with Y ~ N(0, n-^a^Ik) and S ~ Then {or0,[V) log{\-MV)) \ \ {n-l)k nk + 2L J (2.2.12) where L Poisson (3^ ELi and V\L ~ Beta [\{k + 2L), i(n l)k). Proof . Let T = n ^JLj ^^nd G = T + 5 so that V w T/S 1 + W 1 + T/S G'

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20 Then eo,i(5) = GxI;,{V) and e^^,(r.5) = GV'a(V) where t*(V) i'aiV) = • estimator GV'(y) of one has (n-l)fc and = E,,,,,[Gtl;{V)-logG-log^{V)-l] (2.2.13) If we write t] = {2a^)~^ Ilf=i then, under the reparametrization (&/cr, 1), 5 and T are independently distributed with S ~ X(n-i)fc ^^"^ ^ ~ Xi(^)Introducing the dummy variable L ~ Poissonirj), V and G are independent conditional on L with 7|I ~ Beta + 2L), i(n l)fc) and G|L ~ xlk+2LThen, G(l V-),{V)) = -£'«/. 2, E,,,.[L(eoAS),a') L{el,,,{Y , S),a')] = E[{nk-l+2L')Al.J where E {n-l)k nk-l+ 2L* ] L' (2.2.14) (2.2.15)

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21 L' ~ Pmsson {^ZLii^^ ^)') and V\L* ~ Beta I + 21*), i(n Ufc) and k-a+l , , (n-l)li ^•^ ^ 2 t;-i^(l-t;)^ The proof is omitted because of its similarity to the proof of the previous theorem. 2.3 Interval Estimation of This section is devoted to the comparison of different confidence intervals for abased on the hierarchical models given in Section 2.2. Using 5 alone, the minimum length confidence interval for with confidence coefficient 1 — a is given by Ca(5) = {(7^ : ci5 < a' < C25}, (2.3.1) where Ci ( > 0 ) and C2 ( > ci ) are obtained from /(n-i)fc+4 ^— j = /(n-i)fc+4 and y^"' /(„_i,fc(x)(ii = 1 a, (2.3.2) jv[x) being the pdf of x^, a chi-sqaure variable with v degrees of freedom ( Tate and Klett, 1959; Maatta and Casella, 1990 ). On the other hand, the shortest unbiased confidence interval for cr^ with confidence coefficient 1 — a is given by Ct{S) = {a' -.dxS 0 ) and ^2 ( > ) are obtained from /(„_i)fc+2 (— j = /(„-i)fc+2 ( j-j and /(„_i)fc(a;)rfi = 1 a. (2.3.4) We now find HPD credible intervals for (y~ under the hierarchical priors considered in (I) (III) and (I)' (III)'. We shall show that a subclass of such HPD credible

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22 intervals have smaller length as well as greater coverage probability (in the frequentist sense) than the intervals described in (2.3.1) and (2.3.3). Consider first the class of priors given in (I) (III). Also, we confine attention to intervals of the form C3{W, S) = {a^ : ai{W)S < a' < a2{W)S}, (2.3.5) where ai ( > 0 ) and 02 ( > ai ) are arbitrary constants, while 0 < (j){W) < 1. Recall that = r/5 and r = n y;'We first find within the class of all intervals of the form (2.3.5) the optimal o{W) which leads to the HPD credible interval based on the prior given in (I) (III). Specifically, the following theorem is proved. Theorem 2.3.1 Consider the class of priors given in (I) (III) with {n — l)k + b — a > 0 and 6 < k + 2. Then, within the cleiss of intervads of the form (2.3.5), the optimal o{w), where (i)o{w) is a solution of /(n-l)fc+6-a+4 { T"} ^ ) ( ^ ) = /(„-i)fc+fc-a+4 ( — — T — ) -Pfc-6+2 (—7 — ; — ) , (2.3.6) where f^{x) denotes the pdf of xl, and F^{x) = P{xl < x). Proof . As before, write R = {(7^)-\ Then from (I) (III), the joint posterior pdf of R and U = A/(n + A) given Y = y and 5 = s is /(r, u\y,s) oc exp[-^r(3 + ut)]r^^''''-''U^''''-''\ (2.3.7) Now integrating with respect to u in (2.3.7), one gets f{r\y,s) cc eip(-ir5)r^«"-i)''+^-»»-ip(xL6+2 < ^'•) = expi-'^r3)M"-'^''^'-'^-'P{xl_,^, < wrs). (2.3.8)

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23 Next we need to show that f{r\y,s} is log-concave. Since exp{—^rs)r^^^"~^^'''^''~''^~^ is log-concave, it suffices to show that P{xi-b+2 — ^'') log-concave. This fact is an immediate consequence of Lemma 1.3.2. Since the posterior pdf of R given Y = y and 5 = 5 is log-concave, using (2.3.8) the HPD credible interval for Rs within the class of all intervals of the form (2.3.5) is given by ( , ) where ^w) is obtained by maximizing /J^ exp(-l^).^(("-^>''+^-»)-p(xU^3 < ^^)dz (2.3.9) with respect to (i>{w). This leads to the solution which is equivalent to (2.3.6) since /^(x) denotes the pdf of xlThis proves the theorem. Next we improve on confidence intervals of the form (2.3.1) and (2.3.2) in a frequentist sense using a subclass of HPD credible intervals described in (2.3.5). Denote the interval given in (2.3.1) and (2.3.2) as Jml, and let denote an interval of the form {cr^ : ci(f>{w)s < < C2iw)s}, (2.3.11) where Ci and are determined from (2.3.2). Also, we denote by L{I) the length of an interval I. We consider the special case b = a { < k + 2 ). If we now set aj = Ci and 03 = cj, from Theorem 2.3.1, we have found a class of HPD credible intervals of the form {a^ : Cioiw)3 < cr^ < CiM^M (2.3.12)

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24 for cr^ where 0o('w) satisfies = (— 1— (-^^ (2.3.13) We now prove the following theorem. Theorem 2.3.2 For an interval given in (2.3.12) and (2.3.13) with 0 < M^) < 1, L{I^) < LUml) ^nd P^2{ P.= (cr= G Iml) for all a\ Proof . We appeal to Theorem 3.1 of Kubokawa (1991) in the special case of chi-squared distribution. By definition L{I^) < L{Iml)To prove that /^^ hM at least as large a confidence coefficient as /ml, following Kubokawa (1991), we need to verify (i) 0o(if) is t in tu and lim (i>o(w) = 1, Ml— too First we prove (i). Making u; — oo, we get from (2.3.13) /(n-l)fc+4 [-77. r7~U~ I ~ f{n-\)k+i \ JT. , , ) (2.3.14) \(lim^^oo 'Po(u'))ci/ V(lim^^oo '?f'o(tf))c2/ Since /(n-i)fc+4(^) = /(n-i)fc+4(^), and / is a chi-squared pdf, we must have lim = 1. Ill— »oo Next to show that Mw) is t in w, differentiate both sides of (2.3.13) with respect to w, and get <^o{w){A + wB) = Mw)B, (2.3.15)

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25 where A = A{w) = -/('„_i)fe+4 C2 k-a+2 1 ~~-^('n-l)fc+4 0+2 (2.3.16) and B = B{w) = — /(„_i)fc+4 ( ) /fc-a+2 ( T~" ) ( 4>0Cl ) '^'^ "'''^ ( (f>QCi I /(n-l)fc+4 (2.3.17) To show that S > 0, we have from (2.3.13) ^ = (^) (^) W /fc-a+2 (5^) IW /fc-a to > 0 (2.3.18) since x/„(z)/F„(x) is J. in x for every v by Lemma 1.3.3, and 0 < — < — . Now, using (2.3.16) and (2.3.13), ^ — /(n-l)fc+4 X(^0 > 0 tw fc-o+2 1 -^/n-lJM^U^) 1 ^('n-l)fc+4(5;^) (n-l)fc+4V,j,0Ci ^0C2 /(„-l)fc+4(^) <^0Cl /(n-l)fc+4(^) (2.3.19) since x/^(x)/^(x) = 2 xj is i in x for every i/, and 0 < ^ < ^. It follows from (2.3.15) (2.3.19) that 4>'^{w) > 0, i.e.
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26 /(n-l)fc+4 { — ) > 0, Cl P„i{(T^ G Isu) for all c^^ where (^o is defined in (2.3.20). For the class of priors defined in (I)' (III)', first analogous to Theorem 2.3.1, define a class of confidence intervals of the form (2.3.5), where is a solution of 1 \ „ f w \ /(n-l)fc+6-a+4 ( ^ ) -Pfc-fc+1 ( \(po{w)aiJ \ 1 w = /(„_i)fc+6-a+4 X7-TFk-b+i (2.3.21) \(po{w}a2j \(pQ{w)a2) Such intervals are HPD within the class of intervals given in (2.3.5) under the prior (I)' (III)'. Now putting 6 = a 2 and a < A; + 3, a. = Ci (z = 1,2), (2.3.21) reduces to 1 /(n-l)fc+2 I I ^fc-»+3 [~-] = f{n-l)k+2 ( — I ^fc-a^3 ( ^ ) (2.3.22) , P^2(a^ € Isu) for all a-. Thus, we have produced different classes of HPD intervals which dommate the minimum length or shortest unbiased confidence intervals. Unlike the

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27 intervals of Cohen (1972) and Shorrock (1990) which have the same length, our intervals have smaller length and greater coverage probability. The intervals proposed by Goutis and Casella (1991) are different from ours although they achieve the same objective as ours. However, it is also possible to develop HPD intervals using the formulation of Cohen (1972) and Shorrock (1982, 1990). Under their formulation, the minimum length confidence interval for with confidence coefficient 1 — a is given by C^{S) = {(7^ : ai5 < < (ai + ci)5}, (2.3.23) where ai ( > 0 ) and Ci ( > 0 ) are obtained from /(n-i)fc+4 f— ) = /(n-i)fc+4 { ; ) and /"\ /(„_i)fc(x)£ii = 1 a, (2.3.24) /„(i) being the pdf of x^, a chi-sqaure variable with v degrees of freedom ( see Maatta and Casella, 1990 ). On the other hand, the shortest unbiased confidence interval for (7^ with confidence coefficient 1 — a is given by C^{S) = {<7^ : a^S < (aj + €2)8}, (2.3.25) where 03 ( > 0 ) and C2 ( > 0 ) are obtained from /(n-i)fc+2 (— ) = /(n-i)fc+2 ( — \ — ) and f(r,-i)k[x)dx = 1 a. (2.3.26) We now find a class of HPD credible intervals for <7" under the hierarchical priors considered in (I) (III) and (I)' (HI)'. A subclass of such HPD credible intervals have greater coverage probability (in the frequentist sense) than the intervals described in (2.3.23) and (2.3.25). Consider first the class of priors given in (I) (III). Also, we confine attention to intervals of the form Cs{W,S) = {a^ .o{W)S
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28 where c ( > 0 ) is an arbitrary constant, while 0 < 0 and b < k. Then, within the class of intervals of the form (2.3.27), the optimal 4>{w) is given by o{'w) is a solution of f(n-l)k+b-a+4 ( T ) Fk-b+2 ( "T-; \] = /(n-l)fc-H>-a+4 ( , , s , ) Fk-b+2 \ , , s , ) » (2.3.28) where /„(x) denotes the pdf of xl, and F„{x) = P{xl < x). The proof of this theorem is omitted due to its similarity to the proof of Theorem 2.3.1. Next we improve on confidence intervals of the form (2.3.23) and (2.3.24) in a frequentist sense using a subclass of HPD credible intervals described in (2.3.28). Denote the interval given in (2.3.23) and (2.3.24) as /ml, and let denote an interval of the form {(7^ : {w)s {w) + Ci)s}, (2.3.29) where Ci is determined from (2.3.24). For the special case 6 = a ( < -f2 ), we have, from Theorem 2.3.3. a class of HPD credible intervals of the form {(T^ : Mw)s <<^^ < (M^) + ci)^} (2.3.30)

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29 for where 0o{w) satisfies 1 \ „ ( w = /(n-l)fc+4 ( J-4— ) f-— ^5^) (2.3.31) We now prove the following theorem. Theorem 2.3.4 For an intervad given in (2.3.30) and (2.3.31 ) with 0 < (poiw) < a^, L{I^) = L{Iml) and P,2{a^ G Io) > PA(^^ € Iml) for all a\ Proof. We again appeal to Theorem 3.1 of Kubokawa (1991) in the special case of chi-squared distribution. By definition L^I^,^) = L{Iml)To prove that /^^ has at leMt as large a confidence coefficient as ImLi following Kubokawa (1991), we need to verify (i) (i>o{w) is t in 10 and lim (i)o{w) = ai, (ii) /(„-i)fc+4 (^) Fk (^) > /(„-i)fc+4 (^(J,+cJ (*o(w)+cJFirst we prove (i). Making tx; -+ oo, we get from (2.3.31) ^'--'^'^^ (lim._l ^o(t.)) = (lim._(M-) + c,)) (2.3.32) Since /(„_i)fc+4(^) = /(„_i)fc+4(^^^), and / is a chi-squared pdf, we must have lim (hoiw) = ai. tu— »oo To show that (i>o{w) is t in ly, consider Wi and 7i;2 such that 0 < < 103. Here. (?i>o(ti'i) and 4>q(w2), respectively, maximize and /*^7' /("-l)fc(^)-^fc-a+2(iyiZ)
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30 To show that (Pq{wi) < (i)oiw2), use Lemma 1.3.4 and Lemma 3.1 of Shorrock (1990), which is stated here. Lemma 2.3.1 Let f{x) and g{x) be two unimodal densities and let = (pf maximize j^'^" f{x)dx and = o) "'^^ \4>Q + Ci J ''\(i>0 + ClJ " V*^0 /(n-l)fc+4 (^) > 0, putting m = k a + 2 {a < k + 2), n = k, y = ^ and x = in Lemma 1.3.1. This completes the proof of Theorem 2.3.4. Also, we may note that the class of intervals given in (2.3.30) includes as its members the ones considered by Brewster and Zidek (1974) and Shorrock (1990) for the special case a = b = 2. To improve on intervals given in (2.3.25) and (2.3.26), choose b = a 2 and a < /: + 4 in (2.3.28) to get (^o from A-'-'" (^) = (^) (^) (2-3-33) Then following the line of the proof of Theorem 2.3.4, one can show that P„2{a^ e I Pq is defined in (2.3.31) and Isu denotes the shortest unbiased confidence interval given in (2.3.25) and (2.3.26). For the class of priors defined in (I)' (III)', first analogous to Theorem 2.3.3, define a class of confidence intervals of the form (2.3.27), where (po{w) is a solution

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31 of /(n-i)fc+fe-a+4 (— — rj i -—, — : ) = /(„-l)fc+6-a+4 [-7-, T) ( T"^ ) (2.3.34) Such intervals are HPD within the class of intervals given in (2.3.27) under the prior (I)' (III)'. Now putting 6 = a 2 and a < + 3, (2.3.34) reduces to ^^"-^'""^ {£) ^''-""^ (^) = (^) ^'-"'^ (^) The resulting interval satisfies P„2{a^ 6 /^) > P^^icr^ G hu) for all a^. 2.4 Numerical Results Tables 2.1 and 2.2 provide the numerical computations of the risk improvement °f ^a,a,iO^^^) over the best multiple estimator S/{n l)k of given by (2.2.11) and (2.2.12) for k = 5,10, n = 2, 3, a = 1, 2, 3 along with several sets of values of 9i, 1 = 1,...,/:. Tables 2.3 and 2.4 provide the risk improvement of e* ^(y , 5) over the best multiple estimator S/{nl)k of given by (2.2.14) and (2.2.15) for the same values of k, n, a and i = l,...,k. The numerical computations were done using the mathematical software Mathematica (Wolfram, 1988). It follows from these tables that risk improvement of e^f^{Y, S) over S/{n-l)k can often be quite substantial even for k as small as 5. For fixed k and a, the risk improvement seems to be decreasing in n. It is our conjecture that the subclass of proposed HB estimators where 2 < a < A; + 2 is an admissible class of estimators for the variance under entropy loss. The percent risk improvement of e^^^.^{Y,S) over S/{n-l)k is plotted agamst the non-centrality parameter given by 77 = uiEtiS, / 2a^ for k = 5, n = 2, and

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32 k = 10, n = 2 in Figures 2.1 and 2.2, respectively. The three lines represent the percent risk improvement for a = 1,2,3. To illustrate the results for the HPD credible intervals for a', random samples were generated for a fixed-effects balanced one-way ANOVA model with k = .5, n = 2, and = 1, and two sets of values of Oi. The values of ci and C2 given by (2.3.1) and (2.3.2) were first obtained with confidence coefficient 1 a = 0.95. Then o is the ratio of the length of the HPD credible interval to the length of the minimum length credible interval ImlAlthough the improvement in the coverage probability is not very large, a fairly large reduction in the length is obtained.

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33 Table 2.1. Percent Risk Improvement of ejji(y, S) over S/{n l)k for k = .5 n = 2 n = 3 a=l a = 2 a = 3 a = I a = 2 a = 3 ^. = 0 V I = 1 V I ^. = i Vt di = 2t-l V t -6.6888 0.0000 5.1374 10.0212 11.6014 11.9838 14.5122 13.1113 11.1324 0.7027 0.5051 0.3391 -6.3054 0.0000 4.3272 8.8562 9.6882 9.3339 10.9909 9.1853 7.1180 0.0317 0.0190 0.0104 Table 2.2. Percent Risk Improvement of ^aaiO^ > ^) over S/{n-l)k for k = 10 n = 2 n = 3 a = l a = 2 a = 3 a=l a =2 a=3 ^. = 0 Vi ^. = 1 Vi d. = \ vt ^. = 2z 1 Vz -5.4041 0.0000 4.4901 15.3777 15.3715 14.8772 1.0460 0.8098 0.6095 0.0000 0.0000 0.0000 -4.7753 0.0000 3.7204 11.8702 11.3846 10.5213 0.0915 0.0623 0.0409 0.0000 0.0000 0.0000

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Table 2.3. Percent Risk Improvement of e*^ ^(y , S) over S/{nl)k for k = b n = 2 n = 3 a = l a = 2 a = 3 a = l a = 2 a = 3 ^, = 0 V I ei = i V t (?< = i vi = 2i 1 Vi -7.2112 0.0000 5.0982 -7.2112 0.0000 5.0982 4.0271 7.5384 9.2252 6.1421 4.5709 3.0861 -6.4158 0.0000 3.9909 -6.4158 0.0000 3.9909 3.4787 6.1658 7.0088 2.2548 1.4616 0.8415 Table 2.4. Percent Risk Improvement of e; „ ^(y, 5) over S/{n l)k for k = 10 n = 2 n = 3 a = 1 a = 2 a = 3 a = I a = 2 a = 3 = 0 V I ei = i V z = 2z 1 V i -5.5902 0.0000 4.5481 -5.5902 0.0000 4.5481 13.4105 11.9067 10.2688 0.0077 0.0054 0.0037 -4.8062 0.0000 3.6538 -4.8062 0.0000 3.6538 8.1890 6.8467 5.0832 0.0000 0.0000 0.0000

PAGE 43

36

PAGE 44

37 Table 2.5. Probability of coverage for /^i,, when = 5, n = 2 and 1 — a = 0.95. n = 2 a = I a = 2 a = 3 = 1 Vi ^, = i Vz 0.9500 0.9500 0.9500 0.9510 0.9510 0.9510 Table 2.6. Average value of (^o for when ^= 5, n = 2 cind I — a — 0.95. n = 2 a = I a = 2 a = 3 0.8897 0.9138 0.9353 0.9596 0.9702 0.9796

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CHAPTER 3 ESTIMATION OF THE REGRESSION VARIANCE 3.1 Introduction In this chapter, we address the problem of estimation of a^°(a > 0), where (7is the error variance in a Hnear regression model and introduce two classes of hierarchical Bayes (HB) estimators for the two nested regression models, r = Xi/3i + X2/32 + € (3.1.1) and r = Xi/3i+6. (3.1.2) These HB estimators dominate the best multiple estimator of 5" (5 being the error sum of squares), eo,a(5) = c^S^^ where ^ _.-. r(|(n-p + 2a)) r(i(n-p + 4a))' under the relative squared error loss I(a,a^-) = (a/a2--l)'. (3.1.3) Although the above loss is considered in this chapter, similar results can be obtained by using the entropy loss given in (1.1.2). Denote 0^ = (/3f,/3[) and X = (Xi.Xj) where and /Sj are vectors of pi and unknown parameters and X is the known design matrix of order n x p. Within the class of estimators equivariant under locationscale transformations, the best multiple estimator eo,a(5) is admissible, but within 38

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39 the larger class of estimators equivariant only under a transformation of scale. eo,a(5) is no longer admissible. A class of hierarchical Bayes priors is considered for each of the two regression models. The class considered for the full model is similar to the class of priors given in (I) (III) of Section 2.2. Gelfand and Dey (1988a) provided Stein-type estimators of dominating the best multiple estimator eo,i(5). The best multiple estimator can be derived as the Bayes estimator of under the reference prior 7r(/9i, . . . ,/3p, (7^) a"^ as given in Berger and Bernardo (1991). For the reduced model, Ghosh, Lee and Littell (1990) introduced a modified version of the priors considered here for the estimation of the response function. In Section 3.2, we develop a class of HB estimators for
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40 The hierarchical model is given in (I) (III) below, where R = rr'^. (I) Conditional on B = f3, R = r and A = A. Y" is distributed as Y" ~ N{XI3, r-iJ„); (II) Conditional on = r and A = A, /3 ~ iV(0, (Ar)-i(X''X)-^); (III) R and A are marginally independently distributed, with fnir) oc r":" and /a(A) oc A-5''(1 + Xy^^'^-^l Writing U = we have fuiu) oc u":''. Then, (i) conditional on f3, S = s, R = r and U = u, (3^N ((1 u))3, r-^l u) (X^X)-') ; (ii) conditional on /3, 5 = 3 ajid U = u, R ~ Gamma (5 + ut) , ^(n — a + 2)^ where t=^{X^X)0(iii) conditional on /3 and 5 = 3 the pdf of U is given by fu{u\y) oc uJ(''-''>(l +uti;)-^("-"+''/(o,i)(u). (3.2.1) Now, under the loss (3.1.3), the Bayes estimator of R" = a'^" is given by EiR" \P,3)/E{R^'' \^,s). From(i)(iii), we have r(i(n-o + 2a + 2)) , I ^' ^) = -7777 —TT^E (5 + UT)-1/3, 5 r(i(n-a + 2)j ^ J

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41 and EiR"" 1/3,5) = r|(n-a + 4a + 2) 'ri(n-a + 2) £; (5 + ar)-'"|/3,5 Under the loss (3.1.3), the hierarchical Bayes estimator of cr^" is given by E{R-°'\^,S) 'a,h, E{R-^<^\^,S) ri(n a + 2a + 2) ^ [(1 + ^W)-"|3, 5 2 7 ri(n-a + 4a + 2)£[(l + tW)-2«|^,5 = 5"° 0a,6,a(u'), say, (3.2.2) where w = t/s. The following version of Theorem 2.1 of Kubokawa (1991) provides sufficient conditions under which e^^^(3,5') dominates the best equivariant estimator eo^a{S) = c^S" of a^" under the loss (3.1.3). Theorem 3.2.1 Suppose ti;— OO (b) a,b,a{^) > ^o,a(u'), wiierc <^o.a(ty) = C exp(-i3).5(-p)-i(/exp{-h)yl-'dy)d3 Tien e^^g,(y3, 5) dominates eo(5) under the loss (3.1.3), i.e., for all (3, a\

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42 Remark . The above inadmissibility result extends to any quadratic loss lQ(a, C7^°) = Q{a'°'){a a'")', Qia'") > 0. Note that -Mx2.2,cM (using (3.2.1) and (3.2.2)). (3.2.3) We will again restrict our attention to the subclass ^aaai^i done in Section 2.2. The next theorem provides conditions on a under which a,a,a(^) satisfies the conditions of Theorem 3.2.1. Theorem 3.2.2 Let 2 < a < p + 2. Then, 4>a,a,a{'w) sa.tis£es conditions (a) and (b) of Theorem 3.2.1. Proof . We first prove condition (b). For every 2 < a < p + 2, using (3.2.1), we simpHfy (l>a,a,a{yj) given in (3.2.2) as (w) = .-g ri(n g + 2a + 2) u'^jl + uti;)-i^"-°+^°^^)c/u ri(n a + 4a + 2) u^{l + uti;)-^<"-»+''«+2)^^ ^ ri(n g + 2a + 2) f^'^'^-^ z^jl z)^("-P^^-^)dn ri(n g + 4a + 2) ^^(l _ ,)i(n-p^4a-2,^„ ''^•^•^^ (_ UW \ Puttmg z = — 1 + UW J

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43 ri(na + 2a + 2)P\Beta{^, ri(na + ia + 2) p '^Betai^^, n^£±^) < w/{l+w) B = 2-p + 2a)P\Beta{^, nz£±l2.) < w/{l+w) p + 4a) p Betai^^, (3.2.5) p [Betal^ -a+2 2 ' n—p+2a \ 2 ' < w/{l + u;) p [Beta(^ -a+2 2 ' n-p+ia \ 2 ' < w/{l +w)] Sea,a,a{w) satisfies part (a) of Theorem 3.2.1 are omitted because of their similarity to the arguments at the end of the proof of Theorem 2.2.2. Exact Risk When a = 1 We will now obtain an exact expression for the risk of e^^i(/3, 5)forO
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44 5 n — a + 4 1 + n — a + 4 ^ ^ n p + 2 ^ n -p + 2 /J' z^(l zjs^^-P+^j^^ n — p + 2 where 2 yi(p-a+2)(^ _ y^i(n-p+2)^^ (3.2.8) (3.2.9) n a + 4 />• 2^(1 _ z)j("-P+2)d^, The following theorem provides an exact expression for the risk difference of eo,i(5) and e^^i(/3,5) under the loss (3.1.3). Theorem 3.2.3 Consider the model under which Y and S are mutually independent with Y ~ N{Xp, a^I„) and 5 ~ cr^xl-pThen for 0 < a < p + 2, = E[{n + 2L){n + 2L + 2)AL,a] (3.2.10) wiiere Al,. = E (2I + a-2)(^„(y)(l 7) f 2 \~V 1 (n-a + 4)(n-p + 2) \n + 2 + 2L n-p + 2j (3.2.11) L ~ Poisson {^(3^X^X(3) and V\L ~ Beta [\{p + 2L), i(n p)) , Proof . Let T = (3'^X^X/3 and G = T + S so that V = l + W l + T/S G'

PAGE 52

45 Then eo,i(5) = GMV) and e^,^i(/3,5) = GM^) ^here u'.(V) ^^i^V) = '^~!^}jXat,i^^ For an estimator Gi^{V) of a", one has = [(Gt/.(K) If ] ^ and n-p+2 (3.2.12) If we write 77 = (2(7^)"^ (5^X'^Xf3 then, under the reparametrization (/3/cr, 1), 5 and r are independently distributed with S ~ Xn-p ^-nd T ~ Xp(^)If ^^e dummy variable L ~ Poisson{T]), then V and G are independent conditional on L with V\L ~ Beta (i(p + 2L), i(n p)) and G\L ~ x^+2LThen, Efi,a,i [(Gt/.(V) 1)' |] = E Ep,,,, \{Gm) 1)' = E (n + 2I)(n + 21 + 2) £;[V»2(y)|L] 2(n + 2L)E[xi^{V)\L] + 1 = E |(n + 2I)(n + 21 + 2) E { (^(K) — J^)' |l) + + 2L + 2 (3.2.13) Then, we evaluate E 2 1 L = E n + 2 + 2L {MV)-UV)} E (n p + 2)2 [2UV) <^^(K) 2(1 -F) (n-p + 2)(n + 2 + 2I) L . (3.2.14)

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46 Differentiating (f>a{V) from (3.2.9) with respect to V, we have p — a + 2 n — p + 2 2V 2(1 V) MV)n — a + 4 2V liV) (n-a + 4)(l -V) n — a + 4 (3.2.15) Substitution of (3.2.15) into the r.h.s. of (3.2.14) yields (n-p + 2)^ (n-a + 4)(l V) n — a + 4 2(1 V) (n p + 2)(n + 2 + 21) Integration by parts gives (3.2.16) E {V{1 vy :{V)\L] = /;v(i-v)^.-^-(i-.)V-v».iV^(^, (P -\-2L n — p' 2 ' 2 [u = (1-u) = (?ia(t^) v=l u=0 1 fp + 2L £±2i_i, 2 ^ ^ 2 P-Ht n-p "I V 2 (1 u) 2 I (?!>a(y)(fT; (3.2.17)

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47 Substitution of (3.2.17) into (3.2.16) yields E (1 V)ct>,{V) /^^^ _y^_ p-a + 2-{n-a + A)V ^ (n-p + 2) ^^ (n-p + 2)2 (n a + 4) n — a + 4 P + 2£, _ , 2(n-p + 2) l 2 ^ ^ (n + 2 + 2L)J = E {\-V)UV) \^_2{v-a + 2) (n-p + 2)2 n-a + 4 (n -p + 2)(rH2 + 2L) n-a + 4^ ' n-a + V '] P + 2L = E (1 V)a{V) r 2 2 (2i: + a-2)(l-7) I (n-p + 2) \n-a + 4 (71 + 2 + 21) (n p + 2)(n a + 4) J = E {2L + a 2){l V)^{V) ( 2 1-V \ (n-p + 2)(n-a + 4) \(n + 2 + 2I) n-p + 2j = A L,a(3.2.18) The theorem follows now from (3.2.13), (3.2.14) and (3.2.18). Remark . It is proved in Ghosh (1992b) that i4o,2 = 0, Ai,2 > 0 for alU = 1, 2, . . ., and Aj,a > 0 for all a 6 (2,p + 2), while Ao.a < 0, for all 0 < a < 2. So, when a = 2 and 7/ = 0, there is no risk improvement over the best multiple estimator of the sample variance.

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48 3.3 HB Estimator for the Reduced Model We consider the following class of hierarchical priors for the reduced model. (I)' Conditional on /3, i/i, iZ = r and A = A, F is distributed as Y" ~ (II)' conditional on i2 = r, Ui and A = A, /3~ iV \ 0 / , (Ar)-^(X^X)-^ (III)' i/i, R and A are marginally independent, with Ui ~ Uniform{W^), fR{r) oc r-5» and /a(A) oc A-5*(1 + A)-5<*-''). Writing U c = X^X = / \ C21 C22 we have the following theorem. Theorem 3.3.1 Under the model given in (I)' (III)', (if conditional on Y = y, R = r and U = u, (3^ N (1 u)/3i + u/3i (^21 C22 + iz;i^22.i (3.3.1; where C22.1 = C22 C21C1/C12;

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(iif conditional on Y = y and U = u, R ~ Gamma ^^(5 + ut), ^(n pi — a + 2)j where t = ^2 ^i2.\^2 (iiif conditional on Y = y the pdf of U is given by fu{u\y) ~ u5('^-'''(l +itti;)-^("-'''-''+'>/(o,i)(") 49 (3.3.2) where w = t/s. Proof . The joint pdf of Y, (3, Ui, R and A is given by f{y,f3,t^i,r,X) oc r" exp -l\\y-X(3\\' x(Ar)2 exp -~ (/3i ^if Cn {f3r i/i) + /3^C22/32 + 2 {(3,u,f €,2^2 X r-t A-5*(l + A)-5(^-*). (3.3.3) Write and (3.3.4) (^1 u^f Cii (/3i «/i) + /3[C22/32 + 2 (/3i i/i)^

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(3.3.5) Integrating with respect to Ui in (3.3.3), and using (3.3.4) and (3.3.5), one gets /(y,/3,i/i,r, A) oc exp — r2(Ar)2 exp (3.3.6) Next observe that y C21 C22 y / \ /3: / r ^ and Cn/3i = Xfy, so that (3.3.7) and, hence, (3.3.8)

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51 Then, substituting (3.3.7) and (3.3.8) in the exponent from (3.3.6), we have (^-(3fc(0-(3) + Xf3^C22.if32 +2 (3: f3^Y Cu (^2 /32) + A/3^C22.i/32 '^x (/3i + C:,'C,202) + Cn'C,2 {/32 -i^j))' X Cn [/3i {(3, + C-^Ci2/3j + Cr/Ci2 (/32 ^2)] +2 + {(^2 '02^ C22 (/32 '^2) + A/3[C22.l/32 = (/3i rjf Cu (^1 T?) + {^2 (32Y C22.X (^2 (^2) + A/3[C22.l/32 where r? =/3i + C-'Ci2/92 -232^22.1/92 + (1 + A)/3^C22.l/32

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52 1 + A Therefore, A -T ~ P2 C^22.lP2 (3.3.9) f{y,rj,P„r,X) a r=T^(Ar)f A-^^(l + A)-^^^-'') xexp (3.3.10) and, hence, / (3, + C-,'Cuf32 /32 AT 0 ° TTa^22.1 /J (3.3.11) Using (3.3.8), one gets (i)' from (3.3.11). Next integrating (3.3.10) with respect to T] and (32, one has /(y,r. A) a exp A 1 + A (1 + A) -2 Put u = A/(1 + A). Then, du = {1 + Xy^dX. Therefore, /(y.^'iA) a exp f { ^ A r (3.3.12) (3.3.13)

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53 and. /(y,r,A) oc + uw)-^^"'"'"'^''^ where w = t/s. This gives us the theorem. Therefore, under the loss (3.1.3), the HB estimator of cr^° is given by E{R°\Y) (3.3.14) E{R^°\Y) \2J TUn ri(n -p^-a + 2a + 2) E[{1 + UWr^^Y] (n pi a + 4a + 2) E [(1 + UWy^^lY] = S° lk,a{^)^ say. (3.3.15) We will again restrict our attention to the subclass e*^_„(y). Using arguments similar to those in Theorems 3.2.1 and 3.2.2, we now have the following theorem. Tiieorem 3.3.2 Under the loss (3.1.3), elaaO^) dominates €q ^(S) for every a € [2,P2 + 2). Note that e* „ i(Y') can be simplified further as S u^(l + uu;)-5("-'''-»+''>(iu \ ^ /o^z^(l-z)^("-P)dz /o^2'^(l-z)5("-''+^'(iz n — pi a + 4 n — pi — a + 4 1 + P2 a + 2 n p + 2

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54 n — p + 2 (3.3.16) where yi(P2-<'+2)(l _ \/)5{n-P+2) (3.3.17) The following theorem provides an exact expression for the risk of e^^^^iY) for o € [0,p2 + 2) under the loss (3.1.3). Tiieorem 3.3.3 Consider the model under which Y and S are mutually independent with Y ~ N{X/3, a^In) and S ~ (t^I.^. Then forO l{V){l V) f 2 l-V (n-a + 4)(n-p + 2) \n + 2 + 2L* n-p + 2 (3.3.19) L* ~ Pm33on (^^2 022.1^2) ^nd V\L' ~ 5e 0 for all / = 1, 2, . . ., and > 0 for all a 6 (2,p + 2), while Ao,a < 0, for all 0 < a < 2.

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55 3.4 Interval Estimation of a' This section is devoted to the comparison of different confidence intervals for £t^ beised on the hierarchical models given in Sections 3.2 and 3.3. Using S alone, the minimum length confidence interval for with confidence coefficient 1 — a is given by Ci(5) = {(7^ : ci5 0 ) and C2 ( > Ci ) are obtained from /„_p+4 [j] = /n-p+4 [j] and 12 fn.p{x)dx = 1 a, (3.4.2) /„(i) being the pdf of x^. a chi-sqaure variable with i/ degrees of freedom (Tate and Klett, 1959; Maatta and Casella, 1990). On the other hand, the shortest unbiased confidence interval for 0 ) and d2 { > di ) are obtained from /n-p+2 = /r.-p+2 (J^) and J2 fn-p{x)dx = 1 a. (3.4.4) We now find a class of HPD credible intervals for under the hierarchical priors considered in (I) (III) and (I)' (III)'. We shall show that a subclass of such HPD credible intervals have smaller length as well as greater coverage probability (in the frequentist sense) than the intervals described in (3.4.1) and (3.4.3). Consider first the class of priors given in (I) (III). Also, we confine attention to intervals of the form C^{W, S) = {c^ : ar(l>{W)S < < a2{W)S}, (3.4.5)

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56 where oi ( > 0 ) and 02 ( > ai ) are arbitrary constants, while 0 < 4>{W) < I. Recall that W = T/S and T = 0^X'^X0. Within the class of all intervals of the form (3.4.5), the optimal (i>o(W) which leads to the HPD credible interval based on the prior given in (I) (III) is given by the following theorem. Theorem 3.4.1 Consider the class of priors given in (I) (III) with n-p + b-a > 0 and b < p + 2. Then, within the cIilss of intervals of the form (3.4.5), the optimal (j!)(iy) is given by Mw), where Mw) is a solution of Jn-p+b-a+4 I "TT ^ ^ p-b+2 (i)o{w)ai J \o(iy)ai/ = fn-p+b-a+A ( , , \ 1 Fp-b+2 ( . ."^^ | » (3-4-6) where f^{x) denotes the pdf of xl, ^.nd Fu{x) = P{xl < x)The proof is omitted because of its similarity to the proof of Theorem 2.3.1. Consider the special case b = a{

Q{w) satisfies f i ^ \f ( ^ /n-p+4 -T', ; — fp-a-\-2 (i)o{w)ci) \o{w)ciJ i . = f,_^^J—^]Fp.^^J-^] (3.4.8) \(pQ{w)C2j \(po{w}C2) Using arguments similar to those given in Theorem 2.3.2, we now have the following theorem. Theorem 3.4.2 For an interval given in (3.4.7) and (3.4.8) with 0 < 4>o{w) < I, L(I^) < L(Iml) and PA P^^ia^ e Iml) for all a\

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To improve on confidence intervals given in (3.4.3) and (3.4.4), choose b = a-2 and a < p + 4 in (3.4.6) to get the following class of HPD credible intervals {a^ : dio is obtained from 1^:*) {^) i^i (^) Following the line of proof of Theorem 2.3.2, it can be shown that L{I^„) < L{Isu), and P^2{(x^ e I^) > P<,s(ct2 g Isu) for all a-, where denotes the HPD credible interval defined in (3.4.9) and (3.2.14) and Isu denotes the shortest unbiased confidence interval given in (3.4.3) and (3.4.4). Next, we improve upon confidence intervals of the forms given in (3.4.1) (3.4.4) using the class of priors given in (I)' (III)'. Using an argument analogous to that given in Theorem 3.4.1, for a < p2 + 2, a class of HPD credible intervals of the form {(T^ : Cio{w)s o{w)s} (3.4.11) for crwhere o{w) satisfies f i ' \f ( ^ \ 4>o{w)ci J \(po{w)Ci/ = f„_^^J^-^]F^.,^J—^] (3.4.12) is obtained. Following arguments similar to those given in Theorem 2.3.2. this class of HPD credible intervals have shorter length and higher probability of coverage than the minimum length confidence interval based on 5 only given in (3.4.1) and (3.4.2). Similarly, for a < p2 + 4, we have a class of HPD credible intervals for cr^ of the form ' ' ^ {(T^ : d^(i>o{w)s < a'^ < d2o{w)s} (3.4.13)

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58 for where (f>o{w) satisfies (^) = {^) {^) <'-^-"' which dominate the shortest unbiased confidence interval based on S only given in (3.4.3) and (3.4.4). 3.5 Numerical Results We now provide some numerical calculations of the percent risk improvement of ^aa,i «a,a.i o^er 5/(n p + 2). Figures 3.1 and 3.2 give the percent risk improvements for n = 10 and p = 3, and for n = 15 and p = 7, respectively, plotted against t] = (2a^)"^ (3^X'^X(3 using the expression given in Theorem 3.2.3. Figures 3.3 and 3.4 give the percent risk improvements for the latter combination with pi = 2 and P2 = 5, and pi = 5 and p2 = 2, respectively, plotted against tj using the expression given in Theorem 3.3.3. When a = 2, our HB estimator is the BrewsterZidek (1974) estimator in the special case of only one cell mean.

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59

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60

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61 Fig 3.3. Percent Risk Improvement of e^^i over for n = 15, pi = 2 and P2 = 5

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CHAPTER 4 ESTIMATION OF THE VARIANCE RATIO 4.1 Introduction This chapter considers estimation of
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64 statistic is (Yi, y'j, 5i, 52), where Yk = 71-^2"^!^, and Sk = 2r=i(^fc. " ^' = 1, 2. Consider estimation oi-q = crlj (t\. If > 6, then under the usual group of locationscale transformations, and the loss L(r,, a) = f1) , (4.2.1) the best equivariant estimator of Tj is Ui + 1 ^2 However, it is shown in Gelfand and Dey (1988b) that under any quadratic loss Q{crl, al) L{t], a), where Q{(tI, cr|) > 0, the best equivariant estimator 60 is an inadmissible estimator of rj, and is dominated by both ^ ' \°' (n,+2)S2 / and (n2-4)5i \ $2 = mm I (nx + i)Err22,;" Certain generalizations of these estimators were also provided by these authors. It should be noted that the estimators 6^ and 62 are analogous to the estimators of Stein (1964) in the one-sample case, where the best equivariant estimator of the normal variance under the group of location-scale transformations was shown to be inadmissible by a class of testimators. As discussed in Section 4.1, such estimators being non-smooth are themselves inadmissible. We now develop two classes of HB estimators of (rl/aj. Let = R^^ {k = 1,2). The first class of hierarchical Bayes priors is as follows: (I) Conditional on Mm = mk, Rk = {k = 1,2) and Ai = Ai, fj, Y2, Si and ^2 are mutually independent with n~Ar(mfc, (nkrk)-') and 5^ ~ r.-^Xn.-i, ^ = 1,2;

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65 (II) Conditional on Rk = and Ai = Ai, Mi and M2 are independent with Ml ~ i"V(0, (AiTi)"^) and M2 ~ Uni for mi — 00,00); (III) Marginally, Ri, Ai and ilj are independent with /Ri(ri) oc r"'"', 0
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66 (Hi) conditional on Yi = yi and S\ = s\, the pdf of U\ is given by fu{ur\yx,s) ~ 4^'-''V + «it^ir'^"^"*'^^"-f(o.i)(«), (4.2.3) w. here wi = ti/si. Proof . Based on (I) (III), the joint distribution is given by fiVi, yi, Si, 32, mi, mj, ri, r2, Aj) oc n{ ' k ^ e 2 5. } X (V,)'"e-V"!n{r-'-)A-'''(n, + A,)-*'-'-' (4.2.4) fc=l ^ ^ Integrating (4.2.4) with respect to mi and m2, one has /(yi, y2, Si, 32, rirj, Ai) ^ / Aj "1 + Ai (ni + Ai) -2 X eip ^1 J ^1 -2 , 2 [ 'ni + Ai Then, from (4.2.5), we have n Gf^) (4.2.5) /(yi, •si, ri, Ai) a rj "1 3 Ai Til + Ai X eip ^1 I ^1 -2 , "1 — — ^yr + •51 2 1 m + Ai' Si (ni + Ai) -2 and /(y2, 52, r-j) oc = e" 5 5, ' (4.2.6) (4.2.7)

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67 Putting Ui = )^i/{ni + Aj in (4.2.6), we have fiVi, ri, ui) a ' u^' exp ^1 f -2 , r^ r _2 . ] nip. 1 ' Now, we have from (4.2.8), / (ri, til yi, si) oc exp ajid, from (4.2.7), / (^2|-S2) OC e~^r-2 This completes the proof of the theorem. Using the loss given in (4.2.1) the HB estimator of rj = Ri is —07 — I ^Oi ,61 ,02 From (ii) of Theorem 4.2.1, one has E {Ri\Yu 50 E (R2' Y2, S2) E (Ri\yu s.) E (R2' Y2, 52) E (R^' Y2, S2) E {R^' Y2, S2) (4.2.8) (4.2.9) (4.2.10) (4.2.11) (4.2.12) E{Ri Vi, s,) 5i E Yu Sy e[r\ Yu 5i) ni ai + 4 E (l + C/iV^i)-2 Yu Sy Si /o^4^'''"'(i + uivrx)-?("^-'"^-"(/ui (4.2.13) Combining (4.2.12) and f4.2.13), the HB estimator given in (4.2.11) of r? simplifies to ^ ^ ^ 5i (n2 a2 3) /o^ 4^''''^'(l + uiH^i)-;<"^-'^+-»dm = ^ <^a,,i,i.aj(H^l) (4.2.14)

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68 The second class of hierarchical priors is as follows: (I)' Conditional on Mk = ruk, Rk = '^fc = 1,2) and A2 = A2, Yi, V2, 5*1 and 52 are mutually independent with Yk ~ jV (rrik, (nfcrfc)"^) and Sk ~ '"fc^Xn^-i. ^ = 1,2; (II) ' Conditional on Rk = rk and A2 = A2, Mi and M2 are independent with Ml ~ Umform{00, 00) and M-i ~ iV(0, (Ajrj)"^) ; (III) ' Marginally, Ri, R2 and A2 are independent with /Ri(n) oc Ti'"*, 0 < ai < ni + 1, /h,(''2) oc rj^"*', 0 < a2 < 712 + 2, and /a,(A2) oc Aj^'^(n2 + A2)-^<'-''> 0<62<3. Let U, = Then /tr,(u2) oc u," Based on the above class of hierarchical priors, the following theorem is now obtained. Theorem 4.2.2 Under the model given in (I)' (III)', one has (i/ conditional on Yk = yk, Sk = Sk, Rk = rk {k = 1,2) ancf U2 = u^, Mi and M2 are mutually independent with Ml ~ N{yu [nin)-') and M2 A^((l -U2)y2, (n2r2)-'(l -U2)) ;

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69 (ii)' conditional on Yk = t/fc, Sk = Sk {k = 1,2) and U2 = uj, -^i and R2 are mutually independent with 11 and Ri ~ Gamma (^-Si, -(ni — ai + l)j , il2 ~ Gamma ( -(^2 + ^^2), ^(^2 — (iii/ conditional on Y2 = ^2 and 52 = 52, tie pdf of i72 is given by * /a(u2|y2, 5) ~ 4^'"'^ V + «2u;2)-'^"'-'''+'^/(o.i)("), (4.2.15) where W2 = t2f 32. ' The proof is omitted because of its similarity to the proof of the previous theorem. Now, from Theorem 4.2.2, the Bayes estimator of 77 under the loss (4.2.1) reduces on simplification to J/ai ,02,63 — E(Rr E {R2' Y2, S2) E (ri E {R2' Y2, S2) £j(l + C/2W^2|V'2, ^2) 51 ("2 ~ ^2 — 2) 52 ("1 ai + 3)' 51 (722 a2 2) ul^'-''\\ + U2H^2)-?<"^-'")rfn2 52 (ni ax + 3) jji 4(i-^)(i + ^^p^,^)-i(n,-a,-2)^^^ • 5i = 4>ai,ai,h,{W2) 02 (4.2.16) In the next section, we derive frequentist properties of the HB estimators 1701,61,02 for certain choices of (ai,6i,a2) and of the HB estimators ^oi,aj,6s for certain choices (01,02,62).

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70 4.3 Risk Dominance over The main focus of this section is to investigate conditions under which some of the HB estimators developed in the previous section dominate 60 = (Si / S2) („2 5) / (ni + 1), the best multipleof 5i / 52 for estimating rj = aj/al. Two general results are derived in this section. The first of these results provides the basis for determining ai, 61 and so that the resulting rja^MA, smaller risk than that of 80 under squared error ( and hence any quadratic ) loss. The second result provides a similar basis for determining aj, aj and 62 so that the resulting ^ai.a2.6, has smaller risk than that of ^0 under squared error loss. In order to prove the first result, consider a class of estimators Si{Si, S2,Wi) = {S1/S2) hC^i) for estimating t/. Denote by /^(i; r) and F^ix; t) the pdf and the distribution function respectively of a chi-square with u degrees of freedom and noncentrality parameter r. Also, to simplify notation, write f^{x; 0) and 0) as f^{x) and Fu{x) respectively. The first theorem of this section is as follows. ^ ; Theorem 4.3.1 Suppose n2 > 6, (a) i{wi) is t in wi, and ' " ^ (b) \\mM^^) = ^^ ' ' ^ ' / ' V Then ^i(5i, 52, Wi) dominates 60 under the loss (4.2.1), i.e., for all mi, al and a\ if (n2-5) /o-^l + zQ-^zr'^^i (i>i{wi) >

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71 = Mwi) (say). Proof . Write n = \nim\/(Tl. Then, foo roo toot X\ 1 ^ = / / / i / "5 / / {(i>\{W\) W /nt-l(xi)/„j-l(x2)Fi(tOiXi;Ti)dxi(fx2 Jo \_UW\ Jo Jo K X2 ) Xi l{wi) 1> Xifni-l{Xi)fn,.i{x2)fl{wiXi\Ti)dXidx2dWi X2 J dwi 2 \ (f>i{wi) 1 > [{wi)—f„^-i{xi)f„,.i{x2)Fi{wiXi-Ti)dxidx2dwi K X2 ) X2 roo roo roof Xj "1 ^ = / / / {'\{wi) I / 2\(i>x{w-i,) U — /ni-i(xi)/„j_i(x2)Fi(iyiXi;Ti)(ixi(ix2
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72 2 { M^i) 1 \ —fni-\{Xl)fni--Lix2)Fi{WiXi;Ti)dXidX2 (4.3.3) Since (t>'i{wi) > 0, the right h 0 if and only if ^Jn,-i{xi)fn,-i{x2)Fr{wrXr;T,)dxidx2 4>iiwi) > Io° Io° ^fni-l{Xl)fn:,-l{X2)Fi{WiXi;Ti)dXidX2 fo° a:i/„i-i(xi)Fi(u;iii; Ti) |/o°° Vn2-i(a;2)«^X2} dxi Io° xlfni-l{Xl)Fi{WiXi; Ti) |/o°° X2^ fn,_i{x2)dx2] dXi ^ (n; 3)"^ /o°°ii/ni-i(ii)Fi(u;iii;ri)(fii ((nj 3)(n2 5))-i /o~ x?/„i-i(ii)Fi(ti;ia;i; Ti)(ia:i _ , Xif„^_i{xi)Fi{wiXi;Ti)dxi 1^ /o°°a;i/ni-i(xi)i^i(t^ia:i;Ti) 0, (4.3.4) where the pdf of Xi is given by /iri(xi) oc xlfr^,-i{xi)Fi{wiXi;Ti) . (4.3.5) For Tj" > r{, ^r"(a;i) Fi{wiXi;t{') T~7 — \ °^ "cw 77 T in lyixi, i.e in ij. /ir'(a;i) Fi(u;iXi;ti') Therefore, E{X^^) is J. in Ti so that (4.3.4) holds for all n > 0 if and only if Xifni-l{Xi)Fi{WiXi)dXi Io° xlfni-i{xi)Fi{wiXi)dxi

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73 ' !^ x\u,-,{x,) h{x,z,)dz,dx, " !^ x\U.,{x,) h{x,z,)dz,dx, = (n2— oj ni-i — ~ ' !-^z\-'(j^e-'^('^^^xfdx,)dz, = ("2-5) — T-rn — ^z±i ; — ' (n2-5) /-^(l + zi)-^z^'jzi ("1 +2) + = Mt^i)(4-3.6) We now investigate conditions under which i7^^i,aj dominates Sq under the loss (4.2.1). Specifically, we have the following theorem. Tiieorem 4.3.2 Suppose > 6. If = 2 and 2 < aj = 6i < 3, then the HB estimator 7;^^^ „j dominates under the loss (4.2.1). Proof . From (4.2.14), putting 02 = 2, 51 (712 5) ul^'-'"\l + tXityO-i("^-°'+-')dui 'S'2(ni -ai + 4)ja^§(i-0^j^^^P^^^_i(„,_a,+6)^^^ _ 01 ("2 5 J Jo \i+^,wj U+mtVi/ (H-u^^y^)^ ""1 52 (ni ai + 4) ,1 / u.w^, / 1 ly, J JO Vl+uiVrJ \l+uiWj *

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74 51 {n,-b) /o^-/(^-^^-'x^(^-°''(l-x)-^<"--^Wx ' 52 (ni ai + 4) x5(i-'»(l x)-5("^+i)(iui 5a (n2 5) P 5e
1 (4.3.9) by appealing to Lemma 2.1.1. To prove that 7701,02,2 dominates 6q, it now suffices to verify conditions (a) and (b) of Theorem 4.3.1.

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75 To verify condition (a), note that Vi = Wi/{1 + Wi), so, as Wi oo, Vi ^ 1. Then, for 02 = 2 and ai = ii < 3, one has from (4.3.7), n2 5 lim oo nj -(1 (4.3.10) To show that 4>i(wi) is t in Wi, define a new pdf, ^(z)ocz5(l-z)i("-^V.,(^). (4-3.11) Then, for 0 < ui < v[, f^>^{z) / f^,{z) oc I^o.v[){z) / I(o,vi){z) T in z. Note that 4>\iw\) i(wi) is t in vi and, hence in wi (see Lemma 2(i), p. 85, Lehmann, 1986). Before deriving an expression for the risk difference of Sq and 77^^ j) note that V^^^ 1 can be written as K 5) Jo^^/(^^^^)x?(^-''0(l x}-^(-^-')du^ S2 (ni ai + 4) x^(i-oi)(l x)5K+i)(iui 51 {n, 5) /o^x'^^'-^^\l x)^("^-^)ciu^ , 52 (ni ai + 4) /j,^^ x5(i-»')(l x)5("'+^)dui ' Si (nz 5) S2 {ni ai + 4) 51 (n2 5) 52 (ni ai + 4) 1 + 1 + /o^^x?(^-°^)(l-x)f("^-^)^m /0^^X5(^-»^»(1 -X)5("^ + I)dlil Vi 3 oi 2 ni + 1 ni + 1 /J^i a;5(i-»i)(l -x)5<"i+i)(iui (n2 5) 5i (ni + 1)52 (l-<^a.(V'l)) (4.3.12) where K{Vi) = (4.3.13)

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76 Now, an expression for the risk difference of 60 and r]"^%^^2 obtained using the following theorem. : , TJieorem 4.3.3 Consider the model under which Yi and Si are mutually independent with Yi ~ N{mi, nf^cr^) and 5*1 ~ c^iXni-i ^2 and ^2 are mutually independent with Y2 ~ iV(m2, V|) and ^2 ~ (^Ixl^-n where > 6. Then for 2 < oi < 3, (^0.--l)^-«1.2.-^-l)^ = E {m + 2Ii)(ni + 2Li + 2) i"' 1\ al,,,, (n2-3)(4.3.14) where >lr,i.oi = E {2Li+ai-2),,{Vi){l-Vi) 2 _ 1-V i (ni ai + 4)(ni + 1) Ui + 2 + 2Li ni + (4.3.15) where Z-i ~ Pot55on *°<^ ^il-^i ~ -^^^o (^(^ + Kj^i 1)). Proof . Let Ti = niFj^ and Gi = Ti + 5i so that ' 1 + H^ l+?\/5i Gi" Then ^0 = GiMVi)/S2 and e^^ = GiVa,(K)/52 where V.(l^i) = ^^^^ V'ax(Vi) = ((na 5)(1 Vi){l a^{Vi))/{ni + 1). For an estimator GMVi) / S2 of T], one has = E mi/
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Next observe that 77 mi /(Fi)/52-l)' EE, m\lai(V^l) 1 52 ni+2Li+2/ V ^2 ni + 2Li+2, E, "»l/(7-l,l,l {i^lW <(^i))' A' -2{MVi) MVi))/S2\Lr — -£'mi/
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78 — 2 Emi/ 0 for all / = 1, 2, . . ., Ai,,ai > 0 for all ci e [2, 3) and Ao^a^ < 0 for 0 < ai < 2. The next theorem is aimed at proving the risk dominance of Vaua^.b^ for certain choices of ai, 02 and 62. The theorem is stated below. Recall that /„(x) denotes the pdf of xl while /„(x; r) denotes the pdf of xt{''')i ^ non-central chi-square with u degrees of freedom and non-centrality parameter r. Theorem 4.3.4 Suppose n2 > 6, (a) (f>2iw2) is I in 11)2, and theorem. (b) lim (f>2{'W2) = w—>oo Then 62(81,82, W2) = 8i4>2{W2) / 82 dominates 60 under the loss (4.2.1 ), i.e., Em,,.l.l {62/v 1)' < [So/v if for all 7712, ^1 and cr^ if M^2) < (^2-4) /o"'(l + Z2)-^4"'^^2 = (f>2o(w2) (sayj.

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Proof . Write = in2m^/(T|. Then, too foo root X\ \^ = 11 {4>2{W2) If a;i/ni-l(a;i)/n2-l(x2)/2(iy2a;2;T2)(iXidl2tit«2 . Jo Jo Jo K X2 ) (4.3.19) Thus, h{w2) 1 X2 I fn^-l{Xl)fn2-l{^2)F2{w2X2\T2)dx-i,dx2 f«>l d Jo I dw2 mXi "I ^ <^2(ty2)— 1| fni-l{xi)fni-l{x2)F2{w2X2;T2)dXidX'2 dwi = l[ X\fni-l{Xl)fni-\ix2)f2{w2X2;T2)dxidx2dw2 Jo Jo Jo K X2 ) m°° ( Xi 1 X\ 2 < (?!>2(l02) 1 \ <^2("'2) — /ni-l(a!l)/n2-l(x2)F2(tW2X2; T2)dxidx2dw2 K X2 ) X2 /•oo roo roof "1 ^ = / / / i<^2(t^2) If a;i/„j_i(ii)/„j_i(x2)/2(it'2a;2;T2)(ixiC^i2ciix;2 70 Jo Jo K X2 ) + f°° f°° f°° ( Xi ) Xi / "^2(^2)/ / 2U2(U^2) l\ —fn,-liXi)fn,-l{x2)F2{w2X2]T2)dXydx2dw2, Jo Jo Jo K X2 ) X2 (4.3.20) so that

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80 2 { (;i'2(w2) 1 \ —fni-l{Xl)fn2-l{^2}F2{w2X2;T2)dXidx2 K X2 ) X2 (4.3.21) Since 4>2{w2) < 0, the right hand side of (4.3.21) is > 0 if and only if Io° fo° ^fn,-l{xi)fn:i-lix2)F2{w2X2;T2)dXidx2 (f>2{w2) < So° Sq° -^fni-l{xi)fni-l{x2)F2{w2X2;T2)dxidx2 /o°° ^2 fn2-l{x2)F2{w2X2] T;) {/o°° Xifn^-l{Xi)dxi} dx2 ^2^frx2-l{X2)F2{w2X2;T2){J^ X J/„j _1 (Xl ) r' hri'{X2) F2{w2X2\t!^) —2(X — t in W2X2, i.e in X2. (4.3.22) (4.3.23) K!,{X2) F2{W2X2\T2) Therefore, E{X2) is t in rj so that (4.3.22) holds for all > 0 if and only if 4>2{'W2) < 1 Jo" ^2^ fn2-\{x2)F2{w2X2)dx2 "1 + 1 !^ ^2^ fn^-\{x2)F2{w2X2)dx2 1 /o°° ^2 Vn.-l(x2) {Sr^ f2{y)dy)dx, "1 + 1 /o°°2;2-Vn.-i(x2) {ir'f2{y)dy)dx2

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81 1 /o°° Jo" /n;-l(j2) f2{x2Z2)dz2dx2 "1 + 1 /o°° IP ^2^ Ui-liXi) f2{x2Z2}dz2dx2 1 /o°° IP e"^3^2 ' e--^{x2Z2)i~'^dz2dx2 Jo Jo e 2 ^2 e 2 (X2Z2)i 'dz2dx2 1 /o"" 4"' (/o~ e-'^('^'^)xT^dx2) dz2 n, + 1 ^u. (^^oc e-^(i+^.)xj^dx2) dz2 1 r 4"'(H^)-'^r(=^Mz2 "2-4 /o"''(l + Z2) ^z] ^dz2 = hoM(4.3.24) We now find conditions under which f}ai%,b2 dominates under the loss (4.2.1). Theorem 4.3.5 If n2 > 6, ai = 2 and 2 < 02 = 62 < 3, tien the HB estimator t}^^ u dominates 60 under the loss (4.2.1). • . , Proof . From (4.2.16), putting ai = 2, * , ^ 5i (n2-a2-2) /o^4^'"'"V+tX2Vr2)-?("^-<''W2 . S2 (ni + 1) J^ul^'-''"\l + u,W2)-'^^-^-'-')du2 ; . '\ (n2 a2 2) /o^ (t^) ^^'""^ (1 + U2P^2)^^"--^^ni£fa^^2 ^^^^'^ /o(iTt^)'^""'\l+"3Vr2)i(-^)^^.„2 51 (712 a2 2) /o^'x?(^-''')(l -x)5("'-^)^x : , • . : .; ^ 52 (ni + 1) j;^ x5(i-»=)(l x)5("=-')(ii • ^ :

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82 5i (n2 5) P Beta (i(3 a2), |(n2-3)) 2 52 (ni + l)p Beta (i(3 a2), |(n2-5)) JX 5^("1 + 1) /o = XJ-^I x)5("'-^)(ix 5i (n2 5) ^ Se 1 (4.3.27) by appealing to Lemma 2.1.1 again. To prove that ^2,02,02 dominates 60, it now suffices to verify conditions (a) and (b) of Theorem 4.3.4. To verify condition (a), note that V2 = 1^2/(1 + ^^2), so, as W2 — >• 00, U2 — * 1Then, for ci = 2 and 2 < 02 = ^2 < 3, one has from (4.3.25), n2 — 5 lim 4>2(w2) = w-2 — »oo ni + 1 (4.3.28)

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83 To show that ^2(1^2) is i in W2, define a new pdf, /„,(z)oc2'(l-2)'("=-%,„,)(z). (4.3.29) < r "i Then, for 0 < t;2 < v'2, /„/ 2{w2) « EiZy^^, where expectation is being taken with respect to the pdf given in (4.3.29), it is J, in V2 and, hence in W2 (see Lemma 2(i), p. 85, Lehmann, 1986). We now provide some numerical calculations of the risk improvement of Tja^fi^^ai over ^0 for certain values of rii and 712For = 6, Figures 4.1 and 4.2 give the percent risk improvement plotted against ti = {2a^)~^niml for ni = 3 and ni = 7 respectively. The figures show that the risk improvement for values of Ti close to zero can be quite substantial even for such small values of ni and 712. The HPD credible intervals for rj could not be provided as they were done in Chapters 2 and 3, since our method of proof, which used the log-concavity property of the chi-square distribution function, could not be carried through in this case as the distribution function of an F distribution does not have this property. Nagata (1989) provides expressions for confidence intervals of rj based on Stein-type testimators.

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Fig 4.1. Percent Risk Improvement of 77a,a,i over for ni = 3 and = 6

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85, 1.5 1.0 0.5 c 0.0 -0.5 1.0 a -2.5 \' ill -1.5 ' ' 1 1 ^ \ 1 L_ 0 10 20 30 40 50 60 Fig 4.2. Percent Risk Improvement of t^^.^.i over 60 for = 7 and = 6

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CHAPTERS ; SUMMARY AND FUTURE RESEARCH In the analysis of linear models, point and interval estimation of the error variance play an important role. For the fixed-effects balanced one-way ANOVA model, we obtained in Chapter 2 two classes of hierarchical Bayes estimators of the variance, subclasses of which were shown to dominate the best multiple estimator of the error sum of squares under the entropy loss. The numerical calculations that were provided in Chapter 2 showed that the risk improvement could be quite substantial at times. Also, our numerical calculations indicate that every member of these two subclasses of hierarchical Bayes estimators is admissible. Such an admissibility study will be an interesting theoretical topic for future research. In Chapter 3, hierarchical Bayes models similar to those proposed in Chapter 2 were considered for two nested regression models to obtain estimators which again dominated the best multiple estimator under any quadratic loss. In both Chapters 2 and 3, highest posterior density (HPD) credible intervals were provided for the error variance. A class of these HPD intervals not only had shorter length but also had higher probability of coverage than the usual minimum length and shortest unbiased confidence intervals. In Chapter 4, two-sample extensions of the hierarchical Bayes models proposed in Chapter 2 were used to obtain two classes of hierarchical Bayes estimators of the variance ratio. A subclass of these estimators was shown to dominate the best multiple estimator of the ratio of the sample variances. . .. • An important topic of future research will be to study the performance of the proposed HB estimators of the error variance in one-way random-effects models. In addition, for such models, estimation of the treatment variance as well as the ratio of

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87 the treatment variance to the error variance are of interest. Klotz, Milton and Zacks (1969) developed Stein-type estimators which dominated the corresponding ANOVA estimators of the variance components, but such estimators being non-smooth cannot be Bayes with respect to any prior under quadratic loss. Portnoy (1971) proposed certain hierarchical Bayes estimators but could not prove any analytical dominance of such estimators over the usual estimators. It may be possible to extend our theoretical findings to the estimation of the variance components or the variance ratio in one-way random-effects ANOVA models. A multivariate extension of our results to estimation of the generalized error variance in multivariate one-way analysis of variance models is also of interest. Shorrock and Zidek (1976), Sinha (1976), and Sinha and Ghosh (1987) proved inadmissibility of the best equivariant estimator of the generalized variance under a variety of losses. Development of estimators of the generalized variance in a multivariate setting using a hierarchical Bayes model remains an open question, and is worthy of future exploration. , , .

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BIBLIOGRAPHY Berger, J. 0. (1985) Statistical Decision Theory and Bayesian Analysis. (2nd edn.). SpringerVerlag, New York. Berger, J. 0. and Bernardo, J. M. (1991). On the development of the reference prior method. To appear in Proceedings of the Fourth Valencia International Meeting on Bayesian Statistics: Dedicated to the Memory of Moms DeGroot. Brewster, J. F. and Zidek, J. V. (1974). Improving on equivariant estimators. The Annals of Statistics, 2, 21-38. Brown, L. D. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. The Annals of Mathematical Statistics, 39, 29-48. Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. IMS Lecture Notes Monograph Series, 9. Cohen, A. (1972). Improved confidence intervals for the variance of a normal distribution. Journal of the American Statistical Association, 67, 382-387. Gelfand, A. E. and Dey, D. K. (1988a). Improved estimation of the disturbance variance in a linear regression model. Journal of Econometrics, 39, 387-395. Gelfand, A. E. and Dey, D. K. (1988b). On the estimation of a variance ratio. Journal of Statistical Planning and Inference, 19, 121-131. Ghosh, M. (1992a). Hierarchical and empirical Bayes multivariate estimation. Current Issues m Statistical Inference: Essays in Honor of D. Basu, Eds. M. Ghosh and P. K. Pathak. IMS Lecture Notes Monograph Series 17, 151-177. Ghosh, M. (1992b). On some Bayesian solutions of the Neyman-Scott problem. Tech. Report No. 407, Dept. of Statistics, University of Florida. Ghosh, M., Lee, L. C. and Littell, R. C. (1990). Empirical and hierarchical Bayes estimation of the response function. Statistics and Decisions, 8, 299-330. Goutis, C. and Casella, G. (1991). Improved invariant confidence intervals for a normal variance. The Annals of Statistics, 19, 2015-2031. Klotz, J. H., Milton, R. C. and Zacks, S. (1969). Mean square efficiency of estimators of variance components. Journal of the American Statistical Association 64, 1383-1402. Kubokawa, T. (1991). A unified approach to improving equivariant estimators. Tech Report No. METR 91-01, Dept. of Mechanical Engineering and Instrumentation Physics, University of Tokyo. 88

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89 Lehmann, E. L. (1986). Testing Statistical Hypotheses (2nd edn.). Wiley, New York. Lindley, D. V. and Smith, A. F. M. (1972). Bayes estimates for the linear model (with discussion). Journal of the Royal Statistical Society B, 34, 1-41. Loh, W. (1986). Improved estimators for ratios of variance components. Journal of the American Statistical Association, 81, 699-702. Maatta, J. M. and Casella, G. (1987). Conditional properties of interval estimators of the normal variance. The Annals of Statistics, 15, 1372-1388. Maatta, J. M. and Casella, G. (1990). Developments in decision-theoretic variance estimation (with discussion). Statistical Science, 5, 90120. Morris, C. N. (1983). Parametric empirical Bayes confidence intervals. Scientific Inference, Data Analysis and Robustness. Eds. G. E. P. Box, T. Leonard and C. F. J. Wu. Academic Press, New York, 25-50. Nagata, Y. (1989). Improvements of interval estimations for the variance and the ratio of two variances. Journal of the Japan Statistical Society, 19, 151-161. Portnoy, S. (1971). Formal Bayes estimation with application to a remdom effects model. The Annab of Mathematical Statistics, 42, 1379-1402. Proskin, H. M. (1985). An admissibility theorem with applications to the estimation of the variance of the normal distribution. Ph.D. dissertation, Dept. of Statistics, Rutgers University. Rao, C. R. (1971). Minimum variance quadratic unbiased estimation of variance components. Journal of Multivariate Analysis, 1, 4:A5-A56. Sacks, J. (1963). Generalized Bayes solutions in estimation problems. The Annals of Mathematical Statistics, 34, 751-768. Shorrock, G. (1982). A minimeix generalized Bayes confidence interval for a normal variance. Ph.D. dissertation. Dept. of Statistics, Rutgers University. Shorrock, G. (1990). Improved confidence intervals for a normeil variance. The Annals of Statistics, 18, 972-980. Shorrock, R. W. and Zidek, J. V. (1976). An improved estimator of the generalized variance. The Annals of Statistics, 4, 629-638. Sinha, B. K. (1976). On improved estimators of the generalized variance. Journal of Multivariate Analysis, 6, 617-625. Sinha, B. K. and Ghosh, M. (1987). Inadmissibility of the best equivariant estimators of the variance-covariance matrix, the precision matrix, and the generalized variance under the entropy loss. Statistics and Decisions, 5, 201-227.

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90 Stein, C. (1956). Inadmissibility of the usual estimator of the mean of a multivariate normal distribution. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1, 197-206. Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Annals of the Institute of Statistical Mathematics, 16, 155-160. Strawderman, W. E. (1971). Proper Bayes minimax estimators of the multivariate normal mean. The Annals of Mathematical Statistics, 42, 385-388. Strawderman, W. E. (1974). Minimajc estimation of powers of the variance of a normal population under squared error loss. The Annals of Statistics, 2, 190-198. Tate, R. F. and Klett, G. W. (1959). Optimal confidence intervals for the variance of a normal distribution. Journal of the American Statistical Association, 54, 674-682. Wolfram, S. (1988). Mathematica. AddisonWesley, Reading, MA.

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BIOGRAPHICAL SKETCH > Sudeep Kundu was born on March 19, 1964 in Calcutta, India. After graduating from high school in 1982, he joined Presidency College, Calcutta. He received a Bachelor of Science degree in April, 1986, with honors in statistics from the University of Calcutta. He then came to the University of Florida to pursue graduate studies in the Department of Statistics. He obtained his Master of Statistics degree in April, 1988, and expects to get his Ph.D. in August, 1992. . ' ' As a graduate student, he was a teaching assistant for three years and worked as a consultant in the Consulting Unit of the Department of Statistics for the last three years. He has been a member of the American Statistical Association since 1990. Upon graduation, he will be working as a Biometrician in the Clinical Biostatistics department of Merck Research Laboratories in New Jersey. *

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Malay Ghosh, Ohairrnan Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. AlanGTAgresti Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. — ^ Ramon C. Littelf Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quaHty, as a dissertation for the degree of Doctor of Philosophy. Pejaver V. Reio Professor of Statistics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sencer Yeralan Associate Professor of Industrial and Systems Engineering This dissertation was submitted to the Graduate Faculty of the Department of Statistics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1992 Dean, Graduate School