THE STEIN PARADOX IN THE PITMAN CLOSENESS
by
Pranab K. Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
and
A.K. Md. Ehsanes Saleh
Department of Mathematics & Statistics
Carleton University, Ottawa, Canada
Institute of Statistics Mimeo Series NO. 1823
April 1987
THE STEIN PARADOX IN THE PITMAN CLOSENESS
By PRANAB KUMAR SEN
I
and A.K.Md. EHSANES SALEH
2
University of North Caorlina at Chapel Hill, and
Carleton University, Ottawa, Canada.
The dominance and related optimality properties of the usual
Stein-rule estimators rest on the adaptation of appropriate quadratic
loss functions.
It is shown that in the light of Pitman-closeness,
the Stein-rule estimators possess a similar dominance property when
such quadratic loss functions are incorporated in the distance
function.
The impact of the Stein phenomenon under the Pitman close-
ness criterion is explored in the finite as well as asymptotic cases,
and some answers to an open query of Rao (1981) are provided.
AMS Subject Classifications:
62C15, 62H12.
(Asymptotic and exact) Pitman closeness;
Key Words and Phrases:
dominance; loss function; maximum likelihood estimator; mean,
median, mode; noncentral chi square distribution; nonparametric
and robust estimators;2risk; shrinkage estimators; Stein rule
estimator; Hotelling T -statistic; U-statistics; Wishart
matrices.
1
2
Work supported by the Office of Naval Research, Contract N00014-83-K-0387.
Work supported by NSERC of Canada, Grant No. A3088.
-2-
1.
Introduction.
For estimating the mean (vector) of a p-variate
normal distribution, under a quadratic loss function, the inadmissibility
of the sample mean (i.e., the classical maximum likelihood estimator
(MLE»
for
was established by Stein (1956).
Later on, James and Stein (1961),
p > 3, constructed a shrinkage estimator which dominates the MLE.
The past twenty-five years have witnessed a phenomenal growth in the
literature on this Stein-rule estimation theory in its diverse tributaries; for some systematic accounts of these related developments, we
may refer to Anderson (1984), Arnold (1981) and Berger (1985), among
others.
Pitman (1937) laid down the foundation of an important concept of
"nearness" or "closeness" of an estimator, and the relationship of this
"Pitman closeness" with other conventional measures of efficiency (of
estimators) has been explored by a number of workers.
However, the
role of the Stein-rule estimation theory in the Pitman closeness has
not yet been assessed fully.
Rao (1981) considered some simple shrink-
age estimators and showed that they need not be the Pitman closest ones.
This led him to the basic query:
Whether a Stein-rule (shrinkage)
estimator is (Pitman-) closer than its usual counterpart (in the
entire parameter space)?
Rao (1981) has argued that the quadratic
loss function places undue emphasis on large deviations which may occur
with small probability, and minimising the mean square error may insure
against large errors in estimator occurring more frequently rather than
providing greater concentration of an estimator in neighbourhoods of
the true value.
This criticism is more justifiable for the Stein-rule
e.
-3-
estimators which, in general, may not have (multi-) normal distributions,
even asymptotically.
Sugiura (1984) has successfully incorporated the
notion of Pitman closeness for deriving improved estimators of the normal
covariance matrix, although his findings are restricted to the
asymptotic case only.
The primary objective of the current study is to focus on the
Pitman closeness as a suitable criterion for discriminating among
competing estimators, and in the light of this, to discuss the appropriateness of the classical Stein-rule estimators.
We shall confine
ourselves to the estimation of the multinormal mean, where the covariance
matrix may be specified, or known up to a multiplicative scalar factor,
or even be totally unknown.
In the light of the Pitman closeness, we
,
shall show that, indeed, the Stein rule leads to improved estimation
in the finite sample as well as in the asymptotic case.
Along with the
preliminary notions, the case of the known covariance matrix is treated
in Section 2.
In Section 3, these findings are extended to the case
where the covariance matrix is of the form
a2 is an unknown parameter.
a2v, where V is known and
The general case of an arbitrary (and
unknown) covariance matrix is considered in Section 4.
The results
obtained in these sections hold for the finite sample size case (i.e.,
they are of exact nature, not asymptotic).
However, these results are
capable of being extended immediately to the asymptotic case and pertain
to a much wider class of shrinkage estimators considered by the current
authors in the past few years.
As such, the concluding section is
devoted to the role of the Pitman closeness in the characterization of
improved estimation in the asymptotic case pertaining to a wider class
of shrinkage estimators.
-4-
2.
Pitman closeness of the James-Stein estimator.
of a parameter
e
An
§
estimator
"
is said to be (Pitman-) closer than another one
e'
if
(2.1)
~,
In the multi-parameter case,
8
some
p
norm
I ~-~ I or more generally a norm
where
~
8'
and
are all p-vectors, for
1, so that in (2.1), we may as well consider the Euclidean
W is a given positive-semi definite (p.s.d.) matrix.
II~-~III'
the Euclidean norm relates to
We say that
e
Thus,
dominates
in the light of the Pitman closeness, with respect to the norm
8'
11'll w
if
(2.3)
with the strict inequality sign holding for at least some
e
E
G.
Now, consider the specific case where
e"
(2.4)
N (e,I).
p -
James and Stein (1961) considered the
Note that here
shrinkage estimator (for
(2.5)
~~S
p
3) :
= {1_c~~~-2}~
and they have shown that
, where Q<c<2(p-2),
"JS
2
Eell ~c -~II I 2.. p, "I e
reduction of the risk for
this context, the choice of
properties.
~
-
close to
~
c
= p-2
We present the following
Q
E
G
(i. e.,
_P
= E""
,
with more
for small
II ~ 11>.
In
is known to have some optimality
e.
-5-
Theorem 2.1.
'"
~
For every
~
p
3
° < c 2 2(p-2) ,
and
"'JS
e-c
dominates
in the light of the Pitman closeness in (2.3).
Proof.
Note that by (2.5),
11~~S_~II2 = II (~_~)_cller2~112
(2.6)
= ,,~_~,,2_c,,~,,-2{2,,~_~,,2+2~'(~_~)_c},
First, consider the special case of
8 = 0.
II 611
Then,
for the central chi square distribution with
p
2
~ ~ , and
degrees of freedom
(OF), it is known [viz., Johnson and Kotz (1970) ] that
(0)
(2.7)
mode = m
P
In fact, here any
8
.
med~an
~
0.
Y
~
c/2 < M(O)
p
(0)
= M
p{211~-~II2+28'(~-~)-c ~ ol~=Q}
Thus,
of
= p-2 <
P
>
~,
= mean, V p
< P
'rJ c/2
would suffice.
We take an orthogonal matrix
2.
~ 2.
m;O) = p-2.
Next, consider the case
A and write
(2.8)
Then
N(Q,!)
and
{211~_~II2
(2.9)
~'(~-~) = 11811Yl.
+ 28'
Therefore,
(~-~)-c}
211 Yl12 +211 ~Ily l-c
= 2{Lf=2
2{~,A
where
2
Xp,A
y~
-
+ (Yl +
~I~II) 2
(t c + A)};
A=
-
(t c + ~1~II2>}
t 11~112,
has the noncentral chi square distribution with
and noncentrality parameter
A (> 0).
p
OF
Thus, it suffices to show that
-6-
for every
p
~
3,
a
(2.10)
P{
the parameter
a
=
q (r;m)
Let
x >
< c 2 2(p-2)
m
e
(~
xp,2 A ~ 2"1 c
-m r
m /r!, r
and every
A
~
0,
+ A} > 1/2.
~
0), and let
be the Poisson probabilities with
0,
gs ()
x
=
{2
s/2/---- }-l -~x ~s-l
s/2
ex, s
~
1,
be the probability density functions for the central chi square
distributions.
Then,
) , th e d ens1ty
.
f unct10n
.
g (A) (x
p
0
2
f
xp, A'
.
1S
given by
(2.11)
Note that
e.
1 { (A)
( A)
}
( A)
(d/dx)gp
(x) = 2" gp_2(x)-gp
(x) , 'rJ x ~ 0, p > 2.
(2.12 )
Thus, if
0.)
m
stands for the mode of
p
2
Xp , A ' we have
(2.13)
and
(A) ( )
g p- 2 x
is
> g(A)
< p
(x)
according as
x
is
inequality rests on the well known unimodality of
Johnson and Kotz (1970); Ch. 28J.
r
~
0, x
(2.14)
~
0,
Writing
g
<
>
(A)
m
p
g (A) ( • )
2 2(x)
p+ r-
=
,
(p-2+2r)x
it follows from (2.13) and some standard steps that
( A)
m
P
< p-2+A, 'rJ A
~
0, p > 2.
( A)
m + ~ p-2+A, 'rJ A ~ 0, p ~ 2.
p 2
As a result, from (2.12) through (2.15), we conclude that
(2.16)
[V1Z.
.
P
In a similar manner, it follows that
(2.15)
the latter
-1
g
2 (x),
p+ r
-7-
Let
G (x)
s
(2.17)
be the survival function corresponding to
G(A) (x)
Note that
P
= p{;/
\
. l' , /\
G (x) - G
s
s-
> x}
-
2(x)
=
s
L >oq(r i A;2)G 2 (x), 'V x
r_
p+ r
2g (x),
s
+
g (x), so that
Vs
~
2, x ~ 0,
~
0, A > O.
and hence,
q(ri A/2) (3/3A) G
2 (P-2+A)}
p+ r
-gp+2r (p-2+A) }
= g~~i(P-2+A)-g~A)
Therefore, we conclude that for every
(2.19)
G(p A)
(p- 2 +A)
p
~
0,
by (2.16).
(~2),
is nonincreasing in
On the other hand, by the asymptotic (in
Johnson and Kotz (1970)
(p-2+A)
A (> 0) .
A) normality of
2
~,A
[viz. ,
J,
(2.20)
1
= - +
2
and
0,
G(O) (p-2+0) = G (p-2) >
p
P
(2.20) ,
Gp (m(O»
p
= 1/2.
Therefore, by (2.19) and
-8-
-G(A) (p-2+')
> 1/2,
1\
P
-
(2.21)
G"(A)
Since
P
c!.. c + A)
> G"(A) (p-2+A), 'rJ 0 <
2
-
' >
f or every
1\
C
P
a•
< 2(p-2),
the proof of
(2.10) is complete.
Note that for the median
for every
in
~
p
1, A > O.
2
of
, we have
~,A
G"(A) (M(A)) =1/2
p
p
On the other hand, the median is sub-additive
A, so that
M(A) < M(O) + A, V A > 0
(2.22)
P
-
P
to choose a sharper bound
and
-
Thus, instead of the upper bound
2(p-2)
(for
p _> 1.
c), it may be possible
2h(p), such that
(2.23)
and (2.10) then holds for every
p-l,
(2.23) holds, although for
=
Thus, for
p
rule [as
p-2
o
< C <
2.
h(p)
~
In particular, for
h(p)
=
p-l + .38, it fails to do so.
2, while there is no shrinkage according to the Stein
= 0],
For
c < 2h(p).
p
it may still be possible to have some, by letting
~
3, this refinement is of very little practical
utility, as under (2.23), (2.10) will be so close to 1/2 that the
improvement in the sense of Pitman-closeness will be hardly noticeable.
3.
,
The case of unknown variance.
As a natural generalization of the
model in (2.4), we now assume that
2
N (e, a V)
P -
(3.1)
where
V
,
is a known (p. d.) matrix and
e
and
a
2
are both unknown.
We also assume that there exist a nonnegative random variable
8
2
and
e.
-9-
a positive integer
(3.2)
m, such that
ms 2/0 2 has the central chi square distribution with
~
on DF, and
and
S2
are independent.
(3.1) and (3.2) hold typically for the usual (univariate) linear models.
For (3.1), it is very natural to employ the usual Mahalanobis distance
I ·11 W
which corresponds to
loss of generality, take
W
V = I
The general case of an arbitrary
section.
Y-1 .
As such, we may, without any
and work with the usual Euclidean norm.
W will be considered in the next
In this case, a Stein-rule estimator (for p > 3) is of the
form
AS
~c
(3.3)
where
c
= c(p,m)
=
{
1 - cS
2 A -2}A
I ~II
~,
satisfies the condition that
0 < c < 2(p-2)m/(m+2)
(3.4)
Proof.
Parallel to (2.6), here, we have
(3.5)
We may set
that
2
Xm
Xm2
and
= rnS
2
/0
2
, so that defining
2
\J,A
as in (2.9), we obtain
2
are mutually independent; without any loss of
X
'1', A
generality, we set
02
verify that for every
= 1.
c
Then, parallel to (2.9), our task is to
satisfying (3.4),
-10-
(3.6)
2
2
}
Q(C,AiP,m) = P {~,A ~ (c/2m)Xm+A
21 ' V A ~
~
where the strict inequality holds for every finite
0,
A (> 0).
Note
that
(3. 7)
f
Q(c, AiP,m)
co
G(A) (A+cy/2m)dG (y).
p
m
a
First, consider the case of
(3.8)
Q(c,AiP,m) =
f
co
(c/2m)
A = O.
Then
{I(p+m)/2t~ (m/21y~P-l(l+y)-~(p+m)dY.
71.t:'/"-
Again, using the unimodality of the density appearing in (3.8) [viz.,
Johnson and Kotz (1970), ch. 24J along with the related mean-medianmode inequality, we obtain that the mode is equal to (p-2)/(m+2»
(3.9)
Q(c,OiP,m) > 1/2,
for
a
and
< c < 2(p-2)m/(m+2).
Further, proceeding as in (2.18), we obtain that
co
(3.10)
(O/oA)Q(c,Aip,m) =
f
g
(A)
(A)
o
p+
P
-2
ta {
2(A+c y/2m)-g
}
(A+cy/2m) dG (y)
m
I
(d/dx) g (Ai (x)
A
/2 }dG (y)
p+
x= +cy m
m
[by (2.12)
J
co
c -1 4m
f { (d/dy)g (A)2(A+Cy/2mJ~G
,lA- (y)
a
m
p+
= 4m {[G (y) ~ (A) (A+ ~)
c
m
dy gp+2
2m
co
=
a
2
(yl{d 2 g(A)2(A + ~) }dy}
m
dy
p+
2m
- JG
a
Jco
4m{ d
-c
dy
(A) (A
gp+2
~)
+ 2m
Iy=O }
I
2
2
4m
c
d
(A)
-G (y) - 2 {-2 gp+2(x) x= A+.£Y. }dy
COm
4m
dx
2m
Jco-
e.
-11-
Note thqt [viz., Johnson and Kotz (1970), Ch. 28J the noncentral density
(A) (
g p+ 2 X
)
( A)
m
> p-2+A > A,
p+2 -
is (strongly) unimodal {with mode
Yp
> 2,
(2.15», and hence,
( A)
(3.11)
(3.12)
(d/dx)gp+2{x) IX=A
2.
0,
Y A 2. 0, p 2. 2,
2
2
(d /dX )g{A ) (x) = (d/dx) {g{A)2{d/dX)
p+ 2
p+
2.
(A)
gp+2{x) (d/dx)
{
[log g{A ) (x)
p+ 2
J}
(A)}
(d/dx) log gP+2{x)
> 0 ,
as
(d/dx) log g
( A)
p+
2{x)
is monotone nondecreasing in
(O/OA)Q{c,A;p,m) ~ 0,
(3.13)
YA
>
x.
Hence,
o.
Finally, note that
(3.14)
. {x.
2
-P-A
\
)
=
p
'1',A
>
1
Q ( c,/\;p,m
h (p+2A) - 12 (p+2A)
+ 1/2
as
and as
A + co,
2{p+2A) + + co,
2
(X
- P - A)/ 12{p+2A)
'1', A
unit variance.
(3.15)
2
m
Xm =
while the left hand side
0
mean and
(3.13) and (3.14), we obtain that
Q(c,A;p,m) ~
J}
-1 2
p > c/2,
is asymptotically normal with
Thus, by (3.9),
(p _£)
A + + co ,
where the last step follows from the fact that
op (I),
[~(":/
-m) 2m "m
21 '
~ A >
o.
by
-12-
Thus (3.6) holds and the proof of the theorem is complete.
Note that here also, the upper bound
be replaced by
p
=
2.
The case of an arbitrary covariance matrix.
N (e,E)
(4.1)
where
p -
-
(for c) may
h(p) < p-l , and this will enable
2h(p)m/(m+2) , for
the shrinkage to be effective, even for
4.
2(p-2)m!(m+2)
We consider the model:
,
E is a p.d. matrix (unknown) and assume that there exists a
p.s.d. stochastic matrix
S
(independent of
has the Wishart distribution
(4.2)
for some positive integer
m (> p).
in (2.2), shrinkage estimators of
"
~),
such that
W (rn,E) ,
p
-
With the quadratic loss function
e,
considered by Berger et ale (1977),
are of the form:
(4.3)
where
p
>
3
and
d =
(4.4)
o
(4.5)
<
C
ch . (WS)
m1n --
,
< c(m,p)i c(m,p) is t in m, lim c(m,p)
2 (p-2) .
Il\-+OO
The dominance of
"S (over
e-c
by Berger et ale (1977).
Theorem 4.1.
dominates
Proof.
8"
For every
" and its minimax character were studied
8)
We have the following
p > 3
and
0 < c
~
2 (p-2)m/(m-p+3) ,
in the light of Pi tman closeness (and the norm
Note that
~
-ll w) .
e.
-13-
Ile-8112 - 2cdT- 2 (e-8) 'ww-ls-le
II ~s -811~
(4.6)
- - w
- - -- - -
+ c2d2T-4§'S-lW-lww-1S-le
-
--
where by the Courant theorem
(4.7)
d
e
for every
E
RP .
-1
,
Thus, to verify (2.3), it suffices to show that
(4.8)
First, consider the case of
Q.
8
Note that the left hand side of
(4.8) is then
(4.9)
where using the spherical transformation, we readily obtain that
-1 2 D 2/ 2
m T = ~ Xm-p+l
( 4.10)
(under
8
0)
[viz., Anderson (1984, p.162)], so that parallel to (3.9), we obtain
that under
Po
8
= Q,
AI
c/2m
For the general case of
~ ~
(4.12)
{
(~-~)
~
-1 A
~ ~
(4.11)
}
V0 < c
> 1/2,
Q,
2{p-2)m/{m-p+3) .
we write
(~_~) 'fl~ = (~_~) ,~-1(~_~) + ~I~-l(~_~)
= (~_~)~-~~~~-l~~~-~(~_~)
where
2
Y - N (O,I)
p - -
and
ms* -W
p
(m,I).
-
+ ~,~-~~~~-l~~~-~{~_~)
Further, we let
-14-
(4.13)
so that.
A is an orthogonal matrix and
(4.14)
Also, we let
Z
= AY
and use the spherical transformation on
Z
Then (4.10) reduces to
(I U
_. ~ 2 + Yl U)
1 V" ;
-
(4.15)
II
U11 2
~ ~2, -l- _
V
II
Y2
"ll\-P+1
Thus, writing
(4.16)
IIul1
2
P
+ ylu l
2
L u.1
+ (VI +Y1/ 2 ) 2
2
, where
2
= X ,:\.- :\.
p
1 2
4" Yl
1 2
:\. = 4"Y = !( e' L-le)
l
4 - - -
e.
we obtain that (4.8) reduces to
(4.17)
so that the proof of theorem 3.1 can readily be adapted to verify that
(4.17) is > 1/2
for every
e
E
~ ,whenever
0
<
c
<
2(p-2)m/(m-p+3),
and this completes the proof of the theorem.
Comparing the range of
c
in Theorem 4.1 with that in Berger et al.
(1977), we observe that we may choose a larger shrinkage factor in our
case.
with
5.
Also, as in Section 3, the result applies to the case of
h(p)
~
p
=2
p-l.
Asymptotic theory.
For underlying distributions not necessarily
(multi-) normal, robust and nonparametric shrinkage estimation theory
has been developed by the current authors in a variety of situations
[viz.,Sen (1984), Sen and Saleh (1985, 1987) and Saleh and Sen (1985,
•
-15-
1986), among othersJ.
Also, for the MLE, the asymptotic theory of
Stein-rule estimation has been studied by Sen (1986b).
An asymptotic
treatment of Pitman closeness of usual BAN estimators has been considered by Sen (1986a).
The current study provides a natural extension
of the latter to Stein-rule BAN estimators.
An essential feature of the asymptotic theory of shrinkage estimation,
studied by these authors, is the proper identification of a (Pitman-)
neighborhood of the pivot where shrinkage is effective; beyond this
o(n-~)
neighborhood, the shrinkage version and the usual unrestricted
version become asymptotically risk-equivalent.
Further, the computation
of the exact risk (or its asymptotic value) of a shrinkage estimator
for non-normal distributions (or non-linear estimators) may become
prohibitively laborious, and may also demand more stringent regularity
conditions (to justify the existence of the limits).
This difficulty
can largely be avoided by incorporating the concept of asymptotic
distributional risk (ADR), and this has also been effectively incorporated in the earlier studies.
Dominance in the light of this ADR
criterion is largely a distributional property, and, generally, may not
require the verification of intricate moment-convergence results needed
for the study of the (asymptotic or exact) risk.
Study of the asymptotic
dominance in the light of Pitman closeness (for Pitman-alternatives)
also involves only the asymptotic distributional results, and hence,
may not require regularity conditions more stringent than those pertaining to the ADR dominance.
Besides, as in the earlier sections, the
Pitman-closeness criterion may require less restrictive conditions on the
shrinkage factor.
-16-
We consider the following model.
{§ ; n > n } be a
Let
-n
-
0
sequence of estimators admitting an asymptotic linear representation
as in Sen (1986a), such that asymptotically
(5.1)
n
~
where
Let
1.:1
A
(6 -6) - N (O,~)
-n p - -
,
is an unknown p.d. matrix (and may even be dependent on
{S ; n > n}
n
-
be a sequence of stochastic matrices, such that
0
S
-n
(5.2)
, in probability, as
~ ~
n
Further, corresponding to the null pivot for
{K}
n
~).
~
00.
6, consider a sequence
of local alternatives
(5.3)
K
n : ~
=
~(n)
Finally, consider a sequence
=n
-1.:1
~,
~ (fixed)
2
{T ; n > n}
n
(5.4)
-
0
_P
E ~.
e.
of test statistics
n > n ,
-
so that under
{K},
n
T
2
0
has asymptotically the noncentral chi-square
n
distribution with pDF and noncentrality parameter
[Alternative forms for
T
2
n
6. = AI ~-lA .
are available for the likelihood ratio test
or some other rank order tests, but these are asymptotically equivalent
A
(under
{K})
n
to (5.4).J
Then, typically a Stein-rule version of
e-n
is given by
•
(5.5)
where
W
is a given p.d. matrix and
of the ADR of
d
n
=
ch . (WS ), n > nO.
IlUn --n
-
In terms
n1.:l(§S_6) [under K J, the dominance results have been
-n n
studied earlier by the authors.
It follows that the proof of Theorem 4.1
-17-
can readily be incorporated (with the further simplification that by
(5.2)
S
can be replaced by
-n
p > 3, under
(5.6)
lim
n~
{K}
n
~)
"-
to show that for every
a
< c < 2(p-2),
in (5.3),
S
p{mll e
-8
I < mil e
-8
I I Kn }
-n -(n) W-n -(n) W
_> 1/2 ,
so that in the light of the (asymptotic) Pitman-closeness (for local
alternatives), the Stein-rule version
{8}.
-n
{§S}
-n
dominates the usual version
This asymptotic dominance result applies to all the shrinkage
estimators considered in the literature for which (5.1)-(5.2) hOld.
In particular, dealing with U-statistics
[as in Sen (1984)J, we
observe that both (5.1) and (5.2) hold (for the latter, we use the
jackknifed variance covariance estimators), so that identifying the
sample covariance matrix as
a special case of U-statistics, we see that
our solution also applies to the case of shrinkage estimators of covariance matrices for possibly non-normal distributions, and this
extends the results of Sugiura (1984) to a wider class of distributions.
This also provides a general answer to the open question of Rao (1981)
in the asymptotic case, while answers for the finite sample case have
been provided in the earlier sections for normal distributions.
The
counter examples of Rao (1981) correspond to a non-normal case for
the variance estimators (for finite sample cases), but in the asymptotic
case, for Pitman alternatives, they would satisfy (5.1) and (5.2), and
hence, the Pitman-closeness hOlds.
-18-
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~,
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!,
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li,
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