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
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Springer
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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
