ON OIARACfERIZATIONS OF PITMAN CLOSENESS
OF SOME SHRINKAGE FSfIMATORS
Debapriya Sengupta
and
Pranab K. Sen
Indian Statistical Institute
203 B. T. Road
Calcutta. India 700035
Depts. of Biostatistics
and Statistics
University of North Carolina
Chapel Hill. NC 27599-3260
ABSTRACf
For the multivariate normal mean model. within a class of
shrinkage estimators (including the so called positive rule
versions). characterizations of Pitman closest property are
studied.
Further. for the same model with the parameter
restricted to a positively homogeneous cone. Pitman closeness of
restricted shrinkage maximum likelihood estimators is established.
1.
INTRODUCfION
For the p-variate normal mean model. under a quadratic loss.
the classical maximum likelihood estimator (MLE) is not admissible
for p
~
3. and shrinkage or Stein-rule versions dominate the MLE
in the sense of having uniformly a smaller (or equal) risk.
Sen.
Kubokawa and Saleh (1989) have shown that a similar dominance
result holds under a (generalized) Pitman closeness criterion
(PaC). and further. for this p
~
2 suffices.
The PaC is an
intrinsic measure of the comparative behavior of two estimators
without requiring the existence of their first or higher order
moments.
On the other hand. a characterization of the Pitman
closest property (PCP) of an estimator (even within a class) may
require an exhaustive pairwise comparisons with other estimators
-1-
-2-
(belonging to the same class), and thereby may generally demand
additional regularity conditions.
Actually, such a PCP
characterization may not universally hold.
The usual transitivity
property which pertains to quadratic or other conventional loss
functions may not hold under the POe [viz., Blyth (1972)], and
hence, some non-standard analys'is (specifically tailored for
specific models) may be necessary for such a pcp characterization.
For some simple (mostly, univariate) models, some success along
this line has been achieved, only very recently, by restricting to
suitable class of equiuariant estimators where equiuariance is
sought with respect to suitable group of transformations which map
the sample space onto itself [viz., Ghosh and Sen (1989), Nayak
(1990) and Sen (1990), among other].
In genuine multivariate
estimation problems, the entire class of equivariant estimators
may be too big to clinch the PCP characterization.
For example,
for the multivariate normal dispersion matrix model, there is a
gradation of various equivariant estimators of the dispersion
matrix, and a generalized) PCP characterization may hold for
certain subclasses, but not for the entire class [viz., Sen, Nayak
and Khattree (1990)].
In this respect, the results in Sen,
Kubokawa and Saleh (1989) are only partial, and there is a need to
incorporate the pcp characterization results for a more
comprehensive account.
Motivated by this query, the first
objective of the present study is to examine the pcp
characterizations of estimators of the multivariate normal mean
vector within a class of shrinkage or Stein-rule estimators.
As
will be seen that for some class of shrinkage estimators, such a
PCP holds while it may not do so for some other class.
Sengupta and Sen (1990) have considered some multivariate
normal mean models when the parameter is restricted to a
positively homogeneous cone, and they have shown that in the light
of the usual quadratic risks, the usual restricted HLE (RMLE)
dominates the unrestricted HLE (UMLE), but is dominated by
appropriate restricted shrinkage MLE (RSMLE).
In view of the
-3-
Pitman closeness results of Sen. Kubokawa and Saleh {1989}. it is
quite natural to inquire how far these PC dominance results filter
through for the restricted parameter space models treated in
Sengupta and Sen {1990}?
This is a dual objective of the current
study.
Section 2 is devoted to the study of PCP characterization of
shrinkage estimators of multivariate normal mean vectors in an
unrestricted setup.
Section 3 deals with the Pitman closeness
dominance of RSMLE over the RMLE when the parameter belongs to a
positively homogeneous cone.
Some general remarks are appended in
the concluding section.
2.
Let X
~ Hp{Q.
PCP OF SHRINKAGE ESTIMATORS
a
2
y}
are unknown parameters and
matrix.
~ 2. Q = {a 1 •...• ap }' and a 2
where p is
y is
a known positive definite {p.d.}
For two estimators 21 and 22 of
matrix g. defining the norm
21 is closer to
Q than
2
1I:s-~IIQ
Q.
and for a given p.d.
{:s-~}' g{:s-~}.
as
we say that
22 {in the norm 1I-lI } in the Pitman sense
g
if
v a.a.
{2.1}
~
with strict inequalities holding for some
Q.
Sen. Kubokawa and
Saleh {1989} considered shrinkage estimators of
2'1'
= ~.-
-2
'P{X.S}SII~IIQ.V
~
where S is distributed as
o ~ 'P{:s.s}
a
a
~
for the form
9-1 Y-1 ~.
{2.2}
~
2 q-1 ~2. independently of X.
q
{p-1}{3p+1}/2p for every {~.S} a.e .• p ~ 2 and
2
, -1 -1 -1
2
II~IIQ. V = ~ y 9 Y ~.
If a is known. in (2.2). S has to be
~
~
~
~
2
replaced by a.
Then. they have shown that
-4-
P...a ,a{lIc5
- ...all Q <
1I~-jlIlQ}
""p
-
~
2
1/2, V (jl, a ),
(2.3)
so that a SMLE ""p
15 dominates the MLE ...
X in the Pitman closeness
measure for all p
~
~(-)
2 and all shrinkage factor
bounds stated above.
An important member of this class of SMLE is
the simple James-Stein (1961) estimator for which
= b : 0
satisfying the
< b < (p-l)(3p+l)/2p,
~(~,S)
where the upper bound is modified in
the light of the PC measure as in Sen, Kubokawa and Saleh (1989).
The PC measure in (2.1) is a pairwise measure.
all 22 €
~
and 21
€~,
for a suitable class
~
If (2.1) holds for
of estimators, then
21 is said to be the Pitman cLosest estimator (PeE) of jl within
the
~
class~.
Let us denote by
satisfies the inequalities 0
~
the class of all SMLE {c5 } where
<P
< ~(~,S)
~
(p-l) (3p+l)/2p.
primarily interested in a characterization of the PeE of
the class
~
(or a suitable subset of it).
We are
...a within
In Sen, Kubokawa and
Saleh (1989), positive rule versions of the SMLE ...
15 , (denoted by
~
...15+)
were also considered, and it was shown that 2 + dominates
in the PC sense.
2~=b
There is a natural question whether a PeE exists
within the class of positive rule versions of SMLE?
We shall
study this problem too.
To set the things in proper perspectives, let us first
consider the simplest situation where a 2 is known, and without any
loss of generality, we may set ...
V = I.
"'P
In this setup, without any
loss of generality, we let ...
Q = "'P
I , so that in (2.1), we need to
use the Euclidean norm II-II and (2.2) reduces to
15
~
=
{I -
~(X,
~
2
2
-2
a ) a IIXII }X.
~
For the usual James-Stein (1961) version, we set
that (2.4) reduces to
(2.4)
~
~
= b (> 0), so
-5-
(2.5)
For the positive-rule version, we set
(2.6)
so that
(2.7)
where a +
= (a
V 0). It follows from Sen, Kubokawa and Saleh
+
(1989) that £ (b) dominates £(b) in the Pitman closeness sense,
for all permissible values of b (> 0) (for which £(b) dominates
in the same sense).
~
This leads us to formulate the following
classes of shrinkage estimators:
~JS
(i)
(ii) ~;S
and (iii)
~
~
= {£(b)
: 0
= {£+(b)
= general
: 0
<b
~
(p-1)(3p+1)/2p}
<b ~
(p-1)(3p+1)/2p}
class in (2.4) with
~ ~
(p-1) (3p+1)/2p.
Even within each of these subclasses. there may not be a Pitman
closest estimator.
2
med(~)
To illustrate this point, we denote by M =
P
and let
~~~)
= {£(b)
: mp
~~~) = {£(b)
~ b ~ (p-1)(3p+1)/2p}
: 0
<b ~
(2.8)
(2.9)
mp }
Then. we have the following.
Theorem 2.1
6(m)
'"
P
within the cLass
Proof.
= '"60
is the Pitman cLosest estimator of 9
#'V
~~~).
Note that
1I£(b) - ~1I2
= II~
_ ~1I2 _ 2b 0211~11-2 ~'(~_~) + b211~11-2
0
4,
-6-
> 0.
for every b
b
~
Therefore. by (2.1). we obtain that for every
mp ' setting a
2
= 1.
Pa {1I20 - ~II ~ 112(b) - ~II}
,..,
2
= Pa{b
,..,
2
- m)
p
>
2(b-m ) X'(X - a)}
p""""""
Panr O~-~) ~ l<b + mp)}
=
,..,
2
1
= PA{~.A ~ A + 2' (mp + b)}:
1
where the last step follows by noting that ,...,
x' (x-a)
,...,,...,
- A and
~
- !
~
1
2
,..,
Hp(!~'
(b + mp )
4)'
2
A = 4" II~II.
= IIx,...,
(2.10)
- 1.
all
2"'"
2
Note that
> mp . for every b > mp .
while by the subadditive property of meA)
p
(2.11)
= median
of
x2r.A~
[viz .•
Sen (1989)]. we have
meA) ~ m(O) + A = m + A.
p
p
p
V A ~ 0.
(2.12)
Hence. from (2.10). (2.11) and (2.12). we have
Pa {1I20 ,..,
~II ~
"2(b) -
~II} ~~. V~.
so that within the class ~J(SI). ,..,
0°
= ,..,oemp )
b
~
(2.13)
mp '
is the Pitman closest
estimator of ,..,
a.
Q.E.D.
~~~). we may
Before we proceed to consider the dual class
note that if in (2.8). we replace mp by any a
~
> 0. then in (2.10) and (2.11). 21 (b
= mp
(and a neighborhood of
for some
+ a) can be made smaller
than m by choosing b sufficiently close to m.
p
-~.
Thus. for a =
p""
Q).
A+
~ (a
+ b) will be less than
°
'"
m~A).
-7-
and hence, (2.13) will be <~.
every
a,
""
Thus, (2.13) can not hold for
and hence, the desired pcp does not hold.
This explains
why the class ~~~) has the vertex £0 and that £0 is the unique
a within
estimator of
""
this class to clinch the desired PC
property.
Let us next examine the pcp within the class
Proceeding as in (2.10), we have for every °
Pa{lI£o -
~~~)
< b < mp ,
~II ~ 1I£{b) - ~II}
""
2 A '"~
= Pi... {"P,
I\.
1 ( m + b );} I\.
' = 4! 11~112
+ 2
.p
(2.14)
Now, we can use (2.24) of Sen, kubokawa and Saleh (1989) and
conclude that £ dominates 2{b) in the Pitman sense whenever
°
1
~
2{mp + b) is
(p-l) {3p+l)/4p, i.e.,
.
where f
p
is
2
~ {p-l){3p+l)/2p - mp = p~
°
<b
!
in p; at p = 2, f
2
1
(2.15)
- fp'
= 0.386 and it monotonically
converges to 0.334 as p increases. In a similar manner, it
follows that on letting fO = (p2_1)/2p - f , 6{bO) is the Pitman
p
p "" p
closest estimator of ~ within the subcalss {2{b); °
(C
~J{S2».
< b ~ b~}
However, 6{bO)
p may not dominate 6° (or any other 6{b),
~
~
~
for b close to m ) in the Pitman sense, and hence, 6{bO) is not
p
""
the Pitman closest estimator within the class ~~~)
p
Similarly, we
shall see that 2° fails to dominate (in the Pitman sense) any
other 2{b) when b is very close (from below) to m.
p
this rests on the following result.
Let M{A) and m{A) be
p
2
respectively the mode and median of "P,A' for A
Lemma 2.1
For every p
~
2 and A
~
The proof of
0,
p
~
° and p
~
2.
-8-
(A) / M{A)
mp
~ p+2
Outline of the proof.
(2.16)
Although (2.16) has been stated in Sen
(1989). the proof remains obscured due to a multitude of
typographical errors.
proof.
As such. we provide here a more direct
Note that as in Sen (1989). we have
(2.17)
as
~~(M~~)
=
g~A)(M~~~).
Vp and A.
Thus. we may proceed as in
(2.23) through (2.28) of Sen (1989) and conclude that for our
(2.16) to hold. it suffices to show that
MX
Writing ~
A
=
(8/8A)M~~ ~
1.
VA
= g{A){M{A»/g{A)(M{A»
p+4 p+2
p+2 p+2'
h
-1\
~
O. p
~
(2.18)
2.
= g{A)(M{A»/g{A){M{A»
p+6 p+2
p+4 p+2
A
(and noting that by the log-concavity of ga )(.). ~
VA
~
0). and making use of (2.13) of Sen (1989). we have
MX = ~ + ~ (1
i.e ..
< ~.
MX{1 - 2A
-
~)[~
(I - ~)} = {I -
< {I
+
MXJ - ~ ~(1
°A
2
-
-
~)
(I - ~)} - (I - ~)
A
~(1 - ~) -
+
2
2A
(I -
~)}.
A
2
~(1 - ~)
(2.19)
-9-
whence (2.18) follows by noting that
~1 - ~) is ~ 1. VA ~ o.
Now. by virtue of the inequalities
(2.20)
we note that for b sufficiently close to mp ' b
A +
1
2'
< m.
p
1
- (m - b)
+ b) = mp + A - 2
p
p
(m
= meA)
p
+ {m + A - meA)} - !(m - b)
p
p
2 p
•
(2.21)
where the right hand side of (2.21) is less than m for A = O. but
P
can be made greater than meA) for moderately large values of A.
p
Hence. although £0 dominates (in the Pitman sense) £(b) (for b
close to m ) near A = O. as A moves away from O. the opposite
p
picture holds.
Hence. within the class
Pitman closest estimator of 9.
~~~). £0 fails to be the
Nevertheless. as A increases.
{em + A - meA»~ - 2! (m - b)}/{2(p + 2A)}~ ~ O. and hence. using
p
p
p
the asymptotic (in A) normality of (X:.A conclude that (2.14)
~ ~
as A
~
(10.
m~A»/{2(P
+
2A)}~.
we
o may not be
Thus. even when {)
'"
strictly closer than £(b). the (generalized) Pitman closeness
measure (distance) is quite small. so that even within the class
~~~). £0 appears to be a natural choice.
can be replaced by
~JS
In this respect.
~~~)
as well.
Let us next consider the case of positive-rule shrinkage
estimators defined by (2.7).
(on letting
a
2
= 1)
For an arbitrary b
> O.
note that
-10-
(2.22)
so that for 0
< b t < b2 , we have
I(II£+(b ) - !!II ~ 1I£+(b2 ) - !!II)
t
= I(II!S1I 2 ~
+ I(II!S1I
2
b t ) + I(b t
< 1I!S1I 2 ~
b2 ) I(!S'!!
~ ~
(1I!S1I
2
- btl)
> b2 ) I(!S'(!S-!!) ~ ~(bt + b2 )),
(2.23)
so that
Pa{II£+(b t ) - !!II ~ 1I£+(b2 ) - !!II}
'"
+ Pa{II!S1I
2
'"
> b2 ; !S'(!S-!!) ~ ~
(2.24)
(b t + b 2 )}·
In order that £+(b ) dominates £+(b2 ), in the Pitman sense, (2.24)
t
should be
~
1/2 for all
a.
In particular at
'"
a = 'V,....,,....,
0, x'a = 0
""
with
probability one, so that the second and third terms in (2.24) drop
out. Thus,
Pa{II£+(b 1) - !!II ~ 1I£+(b2 ) - !!II
'"
la'"
=
O}
'"
(2.25)
which is
!!
~
1/2 only when b 1 is
= Q, Pa{lI£+ (b 1)
'"
b1
~
mp .
- !!II
~
+
~
mp .
11£ (b2 ) - !!II}
In a similar manner, at
~
1/2 for b2
< b1
From this perspective, an ideal choice of b
1
only when
is mp '
i.e., the estimator
(2.26)
-11-
But. then the question is whether the pcp holds for £0+ for the
entire class of positive rule estimators [of the type 2.7)] or for
suitable subclasses? For this. we consider the following:
<e+(1) = {£+ (b) : m
p
JS
+(2)
<eJS
=
<0+
within the
00+( __ o+(m
'"
'"
cLass
<e~~I)
p
~
b
bO
= (p-l)(3p+l)/2p}.
{£+
(b) .. 0 < b
~
.
»
~JS
(2.27)
(2.27)
mp }'
_ <0+(1) U <0+(2)
-
~JS
Theorem 2.2
~
(2.28)
~JS·
~LS the p.~tman c 1. osest
•
est~mator 0
F _.S
"-
Note that for every b > mp • by (2.24).
Proof.
PS{II£O+ - !!"
~ 1I£+(b) - !!"}
'"
=
P!!{1I~1I2 ~ mp } + P!!{mp < 1I~1I2 ~ b. ~'!! ~ ~(1I~1I2 - mp)}
+
PS{II~1I2 > b. ~'!! ~ 1I~1I2 -l<mp + b)}
'"
> IIXII 2
= Pa{X'S
#'V
#'V
'"
-
'"
- 2!(m + b)} + Pa {IIXII
p
,..., '"
2
2
1
~ mp • '"X'S'" < IIXII
- *<2 m +b)}
'"
p
< 1I~1I2 ~ b. 1I~1I2 - ~(mp + b) ~ ~'!! < l<1I~1I2 - mp)}
- Pa{mp
'"
= P!!{~'!!
+ Ps {IIXII
'"
~ 1I~1I2 _ mp } + P!!{1I~1I2 _ l<mp + b) ~ ~'!! < 1I~1I2 - mp }
2
'"
- PS{m
,.., p
= PS{~'!!
,..,
1
~ m • X'S < IIXII 2 - *<2
m + b)}
p
'" '"
'"
2
2
< IIXII
<
b.. .IIXII
,..,
-,
,
p
1
- *<2 mp. + b)
~
x'a
"',..,
~
2
1
*<2 IIXII
- mp )}
,..,
2
2
2
~ IIXII
X'S ~ IIXII
- mp }
a {IIXII ~ mp . "''''
,.., - mp } + P,..,""
'"
-12-
+ PO{m < IIXII 2 ~ b, 2!(IIXII 2 - m ) ~ X'O ~ IIXII
'"
p
'"
'"
p
'" '"
'"
2
- m}
p
+ P!!{1I~1I2 > b, 1I~1I2 _ ~mp + b) ~ ~'!! ~ 1I~1I2 - mp }
~ P!!{~'!! ~ 1I~1I2 _ mp }
= PA{X:,A ~
= PA{~-p,A
mp + A}
(A
=~
2
1I!!1I )
~ meA) + [m + A - meA)]}
p
p
p
~ PA{~
-p,A <
- meA)}
p
(as meA)
p
<
- mp + A,
VA)
1
(2.29)
= 2'
This completes the proof of the theorem.
Theorem 2.3
the class
Proof.
60+ is not the Pitman closest estimator of 0 within
'"
'fl~~2)
Note that for every 0
< b < mp ,
PO{1I20+ - !!II ~ 112+(b) - !!II}
'"
=
Po{II~1I2 ~ b} + PO{b < 1I~1I2 ~ mp ' 1I~1I2 - b > ~'!!}
'"
'"
+ PO{IIXII 2 > m , X' (X - 0) > 2!(m + b)}
'"
'"
+ PO{b < II~II
'"
p
2
'"
~
mp '
'"
'"
II~II
2
-
p
1
- 2 (mp + b)
~ ~'!!
< (II~II 2 - b)/2}.
(2.30)
Now, the first term on the right hand side of (2.30) can be
expressed as
-13-
PA{~'A ~
A+
~
= PA{~-p,A
~ meA}
+ (A + m - meA)} - -21 (m - b)},
p
p
p
p
(mp + b)}
= ~ 1I~1I2)
(A
(2.31)
where
(A)
~
mp
A + mp ' V A
~
0
and
21 (mp - b) ) o.
(2.32)
Consequently, noting that m(O} = m , we conclude that for every
p
p
< mp ' there exists a A, say ~() O}, such that A + ~ (mp + b) ~
b
(A)
mp
, VA
~ ~,
and hence, (2.31)
.
IS ~
1
2'
~~.
VA
Actually, by
virtue of Theorem 1 of Sen, Kubokawa and Saleh (1989), whenever
21 (mp + b)
b
~
~
(p-1}(3p+1}/4p
i.e.,
(p-1+~p)
(P-1}(3p+1}/2p - mp = (p-1}(3p+1}/2p -
= (p-1}(p+1}/2p -
~p
(where ~p ~ 1/3, V p ~ 2
= (p2 - 1}/2p -
If
=
b~, say,
(2.33)
~p! 1/3 as p -+ (I); ~2 = .45),
and
~
(2.33) exceeds 1/2, for all A
only values of b ) b~.
~p
20+
O.
Thus, it suffices to consider
is the Pitman closest estimator of
~ within the class ~;~2}, then (2.30) has to be ~ ~, for every
b
< mp
and A
O.
~
We consider the case of b approaching m from
p
below and denote this by m-.
p
Then the third term on the right
hand side of (2.30) can be made arbitrarily small (and zero in the
limit b = m-).
For the first term, using (2.16) we conclude that
p
whenever A +
~ m~A}}
m~
= 1/2.
)
M~~~ n m~A}}, PA{~'A ~
A+
m~} < ?A{~'A
For this, proceeding as in (2.18}-(2.19), we have
M~~ = p
+
~ =p
+ A - A( 1 -
~},
(2.34)
-14-
where X(1 -
~)
O. V X
+ X. for every X
for every X
~
~
O. and as X increases. X(1 -
~)
Thus. there exists an X (> 0). such that
O
converges to 2.
M~~ < m~
~
is
X '
O
~ XO' so that the first term is < ~.
Further. using the asymptotic (in X) normality
of (x:.X - p - X)/f2(p + 2X)}~
[viz .. Johnson and Kotz (1970)].
we obtain from the above that as X increases.
(2.35)
To complete the picture. we consider the second term of (2.30).
Let Y
= (Y •.... Y )' ~ N (0.1) and let Q = ~~ 2 Y~.
~
p
p ~ ~
J=
J
1
Then we may
rewrite (for b = m-).
p
= P{Q + (Y
1 + X)
2
< mp-
2
+ X . Q + (Y
1 + 2A)
2
~
-
mp }'
(2.36)
Using Anderson's (1955) lemma and some standard arguments. it is
easy to verify that (2.36) is
~
in
X(~
to 1/2 and it converges to 0 as X ~
bounded from above by P{Q + (Y
~
P{Y 1
o at
~
-X} (for X
~
1
0); at X = O. it is equal
Moreover. (2.36) is
00.
+ 2A)
2
~
mp }
~
P{Y
1
~
~
-2X + m }
p
~
mp )' so that as X increases. it converges to
an exponential rate in X.
COmparing this with (2.35). we may
therefore conclude that for large X. (2.30) is strictly smaller
than ~ (albeit very close to 1/2).
£+ (b)
Thus. while 60+ dominates
~
(for b close to mp )' in the PC sense. for small to moderate
values of X. it fails to do so for larger values of X.
Hence. the
PC dominance of 60+ does not hold. and the proof of the theorem is
~
complete.
-15-
Theorems 2.2 and 2.3 imply that within the class
no Pitman closest estimator.
there is
Nevertheless. 00+ emerges as a
'"
strong contender; it has the PCP over the subclass
as a greater domain of
+
~JS
~~~1) as well
~~~2). and even on the complementary part
(of ~~~2». (4.30) is quite close to 1/2. indicating near
attainability of the PCP.
In the above comparison. we may note
that with respect to a quadratic error risk. within the class of
Stein-rule estimators of the form (2.5). an optimal choice of b is
p-2 (=
M~O». for p ~ 3. Note that for p
= 2. p-2 = 0. and hence.
there is no shrinkage factor. so that by the results of Sen.
Kubokawa and Saleh (1989). £0 dominates the Stein-rule estimator.
A natural query is whether such a PC dominance is true for p
The answer is in the negative.
~
3?
Neither £0 nor £(p-2) dominates
the other in the light of the Pitman closeness measure.
To draw
this conclusion. let us first note that £(p-2) belongs to the
class ~(2) [see (2.9)].
JS
have
Pa{II£O - QII
,.,
As such. if we proceed as in (2.14). we
~ 1I£(p-2) - QII}
2
= PA{~.A
Note that mp
~
p-1 - 1/3.
~
~ A+
1
2
p-1 + 1/3 for every p
When A
= ° (or
(2.37)
(mp + p-2)}.
~
2. so that
21 (mp + p-2) is
1
is close to 0). 2 (mp + p-2) + A is
~ meA)
and hence. (2.37) is ~ 1/2. while as A increases.
p
!2 (mp + p-2) + A exceeds M(A)
and by (2 16) meA) / M(A)
p+2"
• P
~
p+2 •
V A ~ 0. so that (2.37) is < 1/2 for large values of A (although
it is very close to 1/2).
£+ (p-2).
A similar picture holds for £0+ vs.
But looking at (2.37) we may gather that for a part of
-16-
the domain of
Q.
2
0
dominates 2(p-2) in the Pitman sense. while
for the complementary part. the superiority of 2(p-2) to
2o
is
From this picture. we may conclude that ,...
6o
very insignificant.
emerges on a good standing relative to the classical James-Stein
estimator too; a similar picture holds for 2
The PCP for general 6
in (2.2) depends heavily on the form
""P
of
~
~.
0+
and in general. a PCP characterization for
(p-1)(3p+1)/2p} may not hold.
~ €
{O < ~(~.s)
Even for the subclass of
estimators of the form
2(b)
= {1
- b SII~II-2}~.
22
(where qs/a ,... ~ . independently of
q
~).
> O.
b
(2.38)
the choice of b depends on
q as well as p. and within this class b = m does not have the
p
PCP.
To see this. we proceed as in (2.10). and obtain by parallel
arguments that for b >
m •
- p
0
Pa a{1I2 -QII
,....
~
112(b) - QII}
2
= PA{~.A
~ A+
1
2 (mp
+ b)q
-1
2
~q}'
(2.39)
If (2.39) has to be greater than (or equal to 1/2. V A
~
O. we
must have
V b
Note that q P
-1
2
~
(2.40)
mp •
2
has the variance ratio (i.e .. F
-)
q
p.q
distribution with DF(p.q). For this F
distribution. the mean
p.q
is q/(q-2). the mode is q(p-2)/p(q-2) and median which is larger
1
than -p mp .
~/~
~
Hence. allowing b to be sufficiently close to m . we
conclude that the median of q
p
2
2
p
q
~ /~
can be made larger than
(mp + b). so that (2.40) does not hold.
!2
A similar treatment holds
-17-
for the analogues of Theorems 2.2 and 2.3. In fact. it is quite
2 2 ) and consider the
intuitive to replace m by m
= med{q ~/~
p
p.q
~
q
estimator 6{mp.q ).
Howevewr. the characterization of the PCP of
this estimator within the entire class of 6{b) in (2.38) (or a
'"
subclass of it) requires more elaborate studies of some properties
of non-central beta distributions. and hence. we shall consider
them in a future communication.
The simple proof of Theorem 2.1
2
or 2.2 or 2.3 may not work out in this case of unknown 0 .
Naturally. the case of arbitrary
~
will be even more complicated
to manipulate properly.
3.
PC DOMINANCE OF RSMLE
As has been mentioned in Section 1. when the parameter
the model X '" HP (9.~»
,...., ,. . ,
'V
~
(for
belongs to a restricted domain (viz ..
positively homogeneous cone). the RMLE fares better than the MLE
and the RSMLE dominates the RMLE in the light of the usual
quadratic error risk.
and Sen (1990).
This has been studied in detail by Sengupta
A parallel picture in the light of the POe will
be depicted here.
For simplicity of presentation. we consider explicitly the
positive orthant modeL for which
(3.1)
Also. as in Section 2. we consider there the model:
X '" H (9.
'"
P'"
0
2
I).
'"
For the case of H (9.
P'"
~). ~
'"
'"
arbitrary (p.d.).
closed expressions for the RMLE and RSMLE are given in Sengupta
and Sen (1990).
For the specific model. ~
= 021.
we have much
more simplified expressions.
= (xl
for
~
V O•...• xp V 0)'.
Then the classical MLE of
confined to 9+. the RMLE is given by
~ is~.
while
-18-
'"
+
9
= X
RM
~
= (Xl V 0 •...• Xp V 0)
~
I
(3.2)
•
For every ~ € RP • let a{~) be the number of coordinates of ~ which
are positive. i.e .•
a{~) = !~=l I{x j > 0).
so that
a{~)
(3.3)
assumes the values O. 1..... p.
Then. the RSMLE of
9. considered by Senputa and Sen (1990). can be simplied as
~
'"
~RSM =
where [a-2]
+
= max[O.
{I -
[a{~) -
a-2]. for a
2]
+
+ -2
II~
II
2
+
u}~.
= O.l ....• p.
the mode is (p-2) and the median is mp (> p-l).
(3.4)
2
~.
Note that for
In view of the
emphasis placed on 60 = 6{mp ) in Section 2. we shall consider.
~
~
side by side. an alternative version of the RSMLE. given by
2
"'*
~RSM = {I - ca{X)u
+ -2 +
II~ II }~.
(3.5)
~
where
(3.6)
Our contention is to compare (3.4) and (3.5) with each other and
with (3.2). in the light of the Pitman closeness criterion.
As in
Sengupta and Sen (1990). we also consider the positive rule
versions:
(3.7)
and compare them with the other versions (under the PaC).
setup. we shall confine ourselves to
~ €
In this
9+.
There is a basic difference in the setup of Sections 2 and 3.
In the unrestricted case. the MLE (X) or its shrinkage version -.p
6
~
is equtuartant under the group of (affine) transformations:
-19-
~ ~
y = ~~,
~
(3.9)
non-singular.
For this reason, we were able to choose
2
ill!=!J= (n, Q'), wheren2= 1I!!1I .
~
in such a way that EY =
Such a canonical reduction may
not be possible for the restricted case; the main difficulty stems
from the fact that the positive orthant 9+ does not remain
invariant under such a non-singular (or even orthogonal)
~,
although scalar transformations on the individual coordinates does
not alter 9+.
In the negation of this equivariance, it is not
surprising to see that the performance characteristics (be it in
the quadratic risk or the PC measure) of the RSMLE and RMLE may
depend not only on 11811 but also on the direction cosines of the
'"
individual elements.
interior of
~
As such, this picture when 8 lies in the
(i.e., !!
'"
> Q)
may not totally agree with the one
when!! lies on the boundary of 9+ (i.e., 8 j = 0 for some j, 1
~
pl.
~
j
However, the relative dominance picture remains the same,
although the extent may differ from the edges to the interior of
9+.
With these remarks, we consider the following.
Theorem 3.1
for p
Proof.
~
For every p
"'* dominates !!RM in the POC, and
2, !!RSM
~
'"
3, !!RSM dominates !!RM in the
Paj.
"'*
We provide a proof for !!RSM
only, as a similar case holds
for !!RSM.
Note that by (3.2),
II~RM - !!1I 2 = II~+ _ !!1I 2
p
2
= ~j=1 {I(X j ~ O)8 j + I{X j
Similarly, by (3.5),
> O){X j
-
2
8j ) }
(3.9)
-20-
A*
II~RSM
- ~II
2
+
= II~
- ~II
2
2
+ ca(X)
4
+-2
a II~
II
'"
(3.10)
As a result.
+
= Pa{(~
,+
-~) ~
1
~
2
a }.
2 ca(X)
'"
(3.11)
'"
where we may note that whenever a(x) = 0 or 1. c ( ) = O. so that
'"
ax
'"
(3.9) and (3.10) are equal.
Hence. we may rewrite (3.11) a little
but more explicitly as
(3.12)
Each of the terms in (3.12) depends on
~
through the individual
a •...• a p ' and therefore a complete working out of (3.12) for a
1
general a (€ 9 ) may require considerable manipulations. For
'"
+
reasons explained after (3.9). we take ~ = (~. Q')'. ~ ~ O. and
for this edge we provide a complete proof.
The simple proof holds
for any of the other p-1 edges. while for higher dimensional
subspaces of 9+, one may require much heavier manipulations.
Dealing with a quadratic risk, some of these manipulations
are reported in Section 6 of Sengupta and Sen (1990), and in view
of the similarity, we shall omit some of these details.
Also. for
simplicity of presentation, we take a = 1.
Let
~(x)
Pa{a(~) =
stand for the standard normal d.f.
0
or
Then note that
I}
'"
= Pa{X
< O}
'" '" - '"
+ ~~ 1 Pa{X.
1=
'"
1
> 0,
X.
J
< 0,
V
j
~ i}
-21-
= ~(-n) 2-(p-1) + ~(n)2-(p-1) + ~(-n)(p-1)2-(p-1)
= 2-(p-1){(p-1) ~(-n) + I}.
(3.13)
Note that at n = O. (3.13) equals to (p+1)/2P . and it
monotonically decreases with n (> 0) with limn~ (3.13) = 2-(p-1).
As such. if p
~
2. (3.12) is
~
1
2'
dominates the RMLE in the POC.
consider the case of p
~
3.
Vn
~
O. and hence. the RSMLE
As such. in the sequel. we only
In this case. for 2
~ a(~) ~
p. we
may identify the two situations:
(i)
Xl and (a-I) of the remaining (p-1) coordinates are
positive. which the rest (i.e .. p-a) negative. and (ii)
Xl and
(p-a-1) of the coordinates are negative and the remaining a are
posi tive.
As such. we can write the second term of (3.12) as
2~Ha+l}}
= 2-(p-1)
[P~l
~
(P-1 ]
2
1
P{Xa •O + ( X1-n)X 1 ~ 2 ca +1 · Xl
a=l
a
> O}
(3.14)
-22-
so that by (3.13) and (3.14). we rewrite (3.12) in the following
way (where c 1
= Co = 0):
(3.15)
Let aa
= G- a (21
2
= P{~a.O
> 21 cal
cal
so that a 1 = 1. aa
*
aa
~
1 Va
2'
As a 1
= 1.
~
2.
for a
and
> 21 for every a
~
2.
* = G- (21 c + ).
a 1
a
* Va ~ 1
aa > aa'
aa
a ~ 1.
and
Then. it suffices to show that for every a
= 1.
~
1.
the proof of (3.16) is simpler. and hence.
we consider only the case of a
2.
~
Note that for
~
= O.
the left
hand side of (3.16) is
-2 aa + 2- aa+ 1
1
*
= 2-(a
a +aa+ 1) = aa
for every a
2.
1
1
~
Letting
~
=
1
* > 2-'
1
1
+ 2-(aa +aa+ 1-2aa )
2~'
(3.17)
we rewrite (3.16) as
(3.18)
The last term on the right hand side of (3.18) is f in
~
(or
~):
-23-
1 (aa+l - aa)
* (> 0) is attaIned
. at
its lower bound. 2
1
*
and its upper asymptote is 2(1
- aa)'
- (21 c + +
I T}O[aa* - G
a 1
a
~
r
2
~ ~
1 in~.
*
< 21 aa'
00) is
= 2'1
is nonnegative
Hence. as
the right hand side of (3.18) converges to a limit
1
*
+ 2(1
- aa)
= 0 (= ~)
The third term.
- (n-x) )] ~(x)dx is
and its upper asymptote (as
~
~
~ ~
00.
* + 0 - 21 aa*
aa
Let us consider the first order (partial)
to~.
derivative of (3.18) with respect
It is equal to
*
1
a -aa ] - g a (-2 c a +1)[~(0)-~(~)]
-~(~)[a
= -(dldx)Ga (x)
where ga (x)
I
g2(x)
= - 21 g2(x)
and g'(x)
a
where g2(x)
= (dldx)ga (x).
= 2l-lkc
e
is
= 2.
For a
! in x. and hence.
(3.19) is bounded from below by
-~
= -~(~)e
On the other hand.
2
1
2(a
2 +
{I - .919} = -.025 ~(-~).
~)
= 0.733.
so that for
(3.18) is bounded from below by 0.733 - .025
.025
[~(TJ)
-
~] ~
0.71
>~.
V TJ
~
O.
(3.20)
~
> O.
Iri ~(y)dy
Also. note that
a
= 2.
= 0.733
~
-
ca + 1 is
-24-
= 2,3,4 and
> a-2 for a
> a-2,
sequel, we consider the case of a
o~
=3
a similar proof works out for a
~~ by {~~)2
We define
o
= a-2
~
third term in (3.19) is
~
and 4.
5.
Hence, in the
1
< a-2).
5 (for which 2 ca +1
~ c a + 1 (> 0), and note that for
-
. 1
2
2
ga{2 ca +1 + ~ - y ) is
~ ~ ~a'
~
g~{~ c a +1) is ~ 0 according as ~ c a +1 is
As such, noting that
~
the opposite inequality holds for a
O.
~
0, V y
~ ~,
so that the
On the other hand, in the last term,
g~{~ ca + 1 + ~2 - (~+x)2) ~ O. V x ~ O. and hence. (3.19) is easily
1
shown to be negative.
~ ~
Since (3.1S) is bounded from below by 2 (as
ro), we complete the proof by showing that (3.19) remains
o
~ ~ ~
negative for all
a (i.e.,
0
~ ~ ~
a =
0
~).
a
For this, we
rewrite (3.19) as
(3.21)
For every
~ ~
Note that
~
*,
a
0, let
for
~
*
~
be defined by 2-1 c ~ 1 + ~2 -
a
= 0,
! in ~, with limW'PO ~*a
=
o.
r-n
is equal to (21 c + ) ~ (< va-2),
and
a 1
~a
= O.
=
*
So
a
*
is
Then, in the last term in (3.21), the
range (O,ro) can be replaced by
~a
*)
(~+~
1
2
g{2 ca +1 + ~
*
(O'~a)'
(~+x»
and further
2
xcp{x)dx
-25-
(3.22)
Finally. using the definitions of
o and
~
a
*•
~
a
it readily follows
that
*
~
a
~ ~
for every
0
a
~ ~ ~
= ~0 .
(3.23)
~
so that from (3.21). (3.22) and (3.23) we conclude that (3.19) is
~ O. V ~ ~ ~o.
a
> 21 at
~
Thus (3.19) is ~ O. V ~ ~ O. Hence. (3.18) being
= 0 and ~ 21 at ~ ~ +00. is ~ 21 for every ~ ~ O. This
completes the proof of the theorem.
We consider next the positive-rule versions in (3.7)-{3.8).
By virtue of (3.5) and (3.8). we have
AMi+ 2
QRSM - Q = -Q I{II~ II
2
< CJ
ca{~»
A*
+ 2
+ {QRSM-Q} I{II~ II ~
2
CJ
ca{~»}'
(3.24)
where (by virtue of
side vanishes for
Co = c 1 = 0)
a{~)
= 0 or 1.
the first term on the right hand
Hence. we have
As in the proof of Theorem 3.1. we consider there only the special
case of Q'
= (~. Q').
~ ~
o. Then the first term on the right of
hand side of (3.25) is given by (3.13). so that (3.25) is
V ~. for p = 2.
CJ
= 1)
For p
~
~
2. the second term is given by (for
1
2'
-26-
(3.26)
Similarly. the last term on the right hand side of (3.25) is given
by
~ {2-(P-l)[(P~I)~ ~(_~)
a=2
p-l
2
2
+ (a-I) f O Ga_l(ca+~ - (x-~) )~(x-~)dx]}
IX)
(3.27)
As such. we may proceed as in (3.18) through (3.23) and conclude
1
2'
~
that (3.25) is
A+
case of
~RSM.
vs.
~
For p
~
3.
~RSM
o. A very similar proof holds for the
Hence. we have the following.
2.
A+
P
~
A
~RSM
Theorem 3.2
V~
A*+
~RSM
dominates
A*
~RSM
in the POC. and for
A
dominates EJRSM in the PCC.
By an adaptation of Theorems 2.1. 2.2 and 2.3. it can also be
shown that there is no PCE of EJ within the class of RSMLE (or
,."
their positive-rule versions) where in (3.5). we allow ca to be
arbitrary.
4.
SOME GENERAL REMARKS
The Theorems presented in Sections 2 and 3 place the POC on a
cOmParable standing with the conventional quadratic risk
criterion.
Moreover. the POC leads to the desired dominance
results even for p = 2. while in the other setup. we are usually
confined to p
~
3.
In this context. we have confined ourselves to
simple shrinkage estimators of the type (2.5) or (3.5).
If
instead of (2.5). we would have considered (2.4). then in (2.10)
(and elsewhere). instead of the constant shrinkage factor b. we
would have a ~(~.
< (p-l) (3p+l)/2p.
0
2
). where
~(.) is arbitrary and 0 < ~(~.
This arbitrariness of
~(.)
0
2
)
eliminates the
possibility of using the simple and direct proof of Theorem 2.2
-27-
(or the others). and a much more complicated approach may be
needed.
Moreover. if the PCE characterization does not hold
within the class ~JS on ~;S' it can not obviously hold for a
larger class generated by such
The results of Sen. Kubokawa
~(.).
and Saleh (1989) can. of course. be used to strengthen the
dominance results of Section 3 to the restricted parameter space
model- however. the PCE characterization will be a trifle harder!
The results presented here are based on the fundamental
properties of (noncentral) chi square distributions.
context with possibly unknown
0
2
In a general
and/or arbitrary ~(.). the
related distributional problems may become untractable.
Moreover.
the role of noncentral chi square distributions may have to be
replaced by that of noncentral beta or variance ratio
distributions.
Some of these properties are under investigation
now and will be reported in a future communication.
Finally. the
results presented here relate to underlying normal distributions.
and they are exact in nature.
In an asymptotic setup (i.e .•
granted the asymptotic normality of an estimator T
"'ll
of
a). the
'"
current results pertain to a wider class. and in that sense. the
results of the last section of Sen. Kubokawa and Saleh (1989)
directly extend to the restricted parametric models in Section 3.
However. we should then keep in mind that like in the case of
quadratic risks. the PC dominance then remains perceptible only in
a local neighborhood of the pivot.
Of course. this is in
conformity with the usual asymptotic setup.
BIBLIOGRAPHY
[1]
Anderson. T.W. (1955).
The integral of a SYmmetric unimodal
function over a SYmmetric convex set and some probability
inequalities.
[2]
Proc. Amer. Math. Soc.
Blyth. C. (1972).
2.
170-176.
Some probability paradoxes in choice from
among random alternatives (With discussion).
Statist. Assoc. 67. 366-373.
]. Amer.
-28-
[3]
DasGupta. S. and Sarkar. s.K. (1984).
concavity.
On TP
2
and log-
In InequaLities in Statistics and ProbabiLity
(ed. Y.L. Tong).
IMS Lecture Notes - Monograph Series. Vol.
5, pp. 54-58.
[4]
Efron, B. (1975).
Adu. in
Biased vs. unbiased estimation.
Math. 16, 259-266.
[5]
Ghosh, M. and Sen, P.K. (1989).
Pitman closeness.
[6]
J. Amer.
Statist. Assoc. 84, 1089-1091.
James. W. and Stein. C. (1961).
loss.
Median unbiasedness and
Estimation with quadratic
Proc. 4th BerkeLey Symp. Math. Statist. Probab.
1.
361-379.
[7]
Johnson. N.L. and Kotz, S. (1970).
Statistics:
Continuous Univariate
Distributions in
distributions~.
John
Wiley, New York.
[8]
Nayak. T. (1990).
Estimation of location and scale
parameters using generalized Pitman nearness criterion.
J.
Statist. PLan. InF. 24. 259-268.
[9]
Pitman, E.J.G. (1937).
parameters.
The closest estimates of statistical
Proc. cambridge PhiL. Soc. 33, 212-222.
[10] Rao, C.R., Keafing, J.P. and Mason. R.L. (1986).
nearness criterion and its determination.
The Pitman
Commun. Statist.
Theor. Math. 15, 3173-3191.
[11] Sen, P.K. (1989).
The mean-median-mode-inequality and
noncentral chi square distributions.
Sankhya, Ser. A 51.
106-114.
[12] Sen, P.K. (1990).
estimators.
The Pitman closeness of some sequential
Sequen. AnaL.
~.
in press.
[13] Sen, P.K., Kubokawa, T. and Saleh, A.K.M.E. (1989).
The
Stein-paradox in the sense of the Pitman measure of
closeness.
Ann. Statist. 17. 1375-1386.
[14] Sen, P.K .. Nayak. T. and Khattree. R. (1990).
Comparison of
equivariant estimators of a dispersion matrix under
generalized Pitman nearness criterion.
(To be published).
-29-
[15] Sengupta. D. and Sen. P.K. (1990).
restricted parameter space.
[16] Stein. C. (1981).
Shrinkage estimation in a
Sankhya. Ser. A. 53. in press.
Estimation of the mean of a multivariate
normal distribution.
Ann. Statist. 2. 1135-1151.