•
,
•
ON SHRINKAGE
R-ESTI~TORS
OF MULTIVARIATE LOCATION
by
Pranab K. Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
and
A,K.Md. Ehsanes Saleh
Carleton University
Ottawa, Canada
Institute of Statistics Mimeo Series No. 1437
•
•
ON SHRINKAGE R-ESTIMATORS OF MULTIVARIATE LOCATION'
by
and A.K.Md. Ehsanes Saleh
Pranab K. Sen
Carleton University
University of North.Carolina
Ottawa, Canada
Chapel Hill, N.C.
Summary.
For a continuous and diagonally symmetric multivariate distrib-
utian, incorporating the idea of preliminary test estimators, a variant
form of the James-Stein type estimation rule is used to formulate some
admissible R-estirnators of location.
Asymptotic admissibility (and
inadmissibility) results pertaining to the proposed and classical
R-estimators are studied.
".
•
1. Introduction.
\
For the mean (vector) of a p-variate normal distribution,
Stein (1956) established the inadmissibility of the maximum likelihood
estimator (the sample mean vector) under a squared error risk measure when
p
~
3,
a simple (non-linear) admissible estimator was later proposed by
James and Stein (1961).
Since then, this theory has been extensively
studied, in the parametric case,
by
a host of workers; a detailed account
of these developments is given by Berger (1980).
The object of the present
investigation is to consider some shrinkage estimators based on rank
statistics wherein the theory of preliminary test estimation (PTE) and
the James-Stein rule estimation are incorporated in the formulation of
the proposed estimators.
AMS Subject Classification: 62C15, 62G05, 62G99
•
Key Words & Phrases: Asymptotic risk, inadmissibility, James-Stein rule,
local alternatives, preliminary test estimators, rank (R-) estimators,
robustness, shrinkage estimators .
'Work partially supported by, the (U.S.) National Heart, Lung and Blood
Institute, Contract NIH-NHLBI-71-2243-L from N.I.H. and partially by the
Natural Sciences and Engineering Research Council of Canada, Grant no A3088.
2.
Let
X. = (X.l,. .• ,X. )', i = l , ... ,n
-~
~
be
~p
n
•
independentand
identically distributed (i.i.d.) random vectors (r.v.) having a p-variate
continuous distribution function (d.f.)
space
EP ,
p ~ 1.
for some
about its location (vector)
(1.1)
where
F6
(~) =
-
F
F
Fe
6 ;
-n
6
, defined on the Euclidean
is assumed to be diagonally symmetric
e(=(6 , ••• ,6 )'),
l
p
(~ -~)
,
~
is diagonally symmetric about
using an estimator
F
(6 1, ... ,6 )'
n
np
E
so that
EP ,
O.
It is desired to estimate
based On (Xl'"'' X ) ,
-
-n
6
where one
considers a quadratic loss function
(1. 2)
for some given positive definite (p.d.) matrix
2.
The risk is then given by
•
Pn (6,6)
= EL(6
,6) = Tr(QV
,.., "'"
. . . . n.......
. . . ...... 0 ) ,
(1. 3)
where
For normal
nE (6
- ,..,.6) (6
- .....
6)' •
"'n
.....0
V'
(1. 4)
-n
F
and for
p
~
3,
it is well known that
is not admissible (in the sense that there exists some James-Stein rule
estimators having smaller risk than
non-normal
F,
the sample mean
X
-n
X
n
for every
,§).
For possible
may not be very robust and may even
be quite inefficient for distributions with heavy tails.
Robust rank
based (R-) estimators of location in multivariate simple linear models
have been studied by Sen and puri (1969), Puri and Sen (1971) and others.
PTE theory for such R-estima tors has also been developed by Saleh and
Sen (1978), Sen and Saleh (1979) and Saleh and Sen (1983), among others.
•
3•
•
The object of the present study is to incorporate the PTE theory in the
proposal of James-Stein rule R-estimators of
~,
and, in this way, some
of the technical difficulties of the James-Stein rule in the nonparametric
case have been avoided.
Along with the preliminary notions. the R-estimators. their PTE
versions and the proposed estimators are considered in Section 2.
The
concepts of asymptotic inadmissibility (for fixed and local alternatives)
are discussed in Section 3. and in the light of these results.
(in-)
admissibility'results (pertaining to the estimators under consideration)
are studied in Section 4.
•
Some general comments are made in the con-
eluding section .
2.
For every real
The proposed estimators.
IXl'J
of
n(~l)
Also, for every
of scores,
whic~
- b
and
I,.... , IXnJ,. - b. I,
for
let
j(=l,. ..• p),
let
b,
a
+
be the rank
i
=
1, .... , n, j = 1, ...... , p ..
.(l) • . . . ,a
nJ
we shall define more formally later on.
+
.(n)
nJ
be a set
Consider the
statistics
(2.1)
T
. (b)
nJ
Then, for monotone
= n
-1 n
L
(i" )
sgn(X ..
i=l
1J
scores,
T
b) a
. (b)
nJ
+
+
nj
(Rij (b»
is
~
in
, 1 " j " p.
b [viz, Puri and Sen
(1971, Ch.6)J, and we set
(2.2)
•
(2.3)
enJ.
= !.2(sU P {b: T
e
-n
,(b) >
nJ
.
=
(e n 1"
OJ + inf{b: T .(b)
nJ
.'
.• ' enp )
<
OJ), 1 " j " Pi
4.
8
_n
is termed an R-estimator of
6.
•
It is a robust, translation-
invariant and consistent estimator having the (coordinate wise) medianunbiasedness property and other desirable properties too.
6
-n
of
would naturally depend on the underlying
fa+. (k), 1
nJ
~ k ~ n; 1 ~ j ~ pl.
F
The performance
and the set of scores
Towards this study, we set for each
j(=l, ... ,p),
a + . (k)
(2.4)
where
or
nJ
U
<
nl
< U
nn
~:(kl)'
l~k~n,
) n+
stand for the ordered r.v. of a sample of size
from the uniform (0,1) d.f. and for every
+
(2.5)
~ . (u)
)
l+U)
= ~j (-2- ;
We also assume the
~j
~ . (u)
)
+
~.
)
u
to be square integrable.
j
be the bivariate marginal d.f. for
v = «v'oj)
-
J~
(0,1),
(1 - u) = 0,
jth marginal d.f. corresponding to the d.f.
F[j£]
€
n
1
~
j
~
p.
Let then
be the
j = 1, ... ,p
F,
for
r£
= 1, ... ,p.
and let
•
We define
by
(2.6)
for
j,£=l, .•. ,p.
Also, we assume that
possesses an absolutely
F[j]
and set
continuous probability density function (pdf)
(2.7)
(2.8)
(2.9)
~J'
•
Jo1 •.) (U) •.) (u)du
,
1
~
j
~
p;
•
5.
•
r
(2.10)
= rlvfl =
«Y ji »'
I
Then, it is known that
n~(e
(2.11)
- iJ.)
-n
-
V
N (O,n.
p--
We consider next the PTE.
e
able when
for
H :
O
~
=
This estimator is particularly suit-
£'
is suspected to be close to
£
so that a preliminary test
is made first, and, depending on the outcome of this test,
Towards this, we define
the ultimate estimator is chosen.
* . 0) )
«mn,
J ...
==
by letting
•
m* .
(2.12)
n, ) i
j,t
for
== n
E. la + . ( R+.. ) a + n ( R.+ n ) 5gn X .. 5gn X-i
-1 n
~=
nJ
~J
n~
n
~~
~J
+
where
== 1,~ .. ,p,
~~
for
R .. (0) ,
~J
i == 1, ... ,n, 1 S j S p.
Then as in Sen and Puri (1967), the test statistic used is
(2.13)
where
(reflexive) generalized inverse of
For small values of
M •
-n
H :
O
(conditionally) distribution-free test for
Ln
the exact (conditional) distribution of
likely sign-inversions of the vectors
n,
L
n
•
~
=
Thus, if
chi square d.f. with
critical value of
Ln
p
2
X (a)
p
£
X., 1 s i s n,
i
n,a
2
n
equally
while, for large
-~
p
degrees of
stands for the upper 100 a%
OF and i f
a
n,
may be based on
generated by the
has closely the chi square distribution with
freedom (OF).
is a
and
T = (T 1"" ,T ) ' , T . = T . (0). 1 ,; j ,; p
-n
n
np
nJ
nJ
point of the
stands for the (conditional)
at the level of significance
a(O < a < 1),
then
6.
(2.14)
as
n
-+
•
IXI.
•
The
PTE
'PT
e
of
il n
may be defined as
Ln
e'PT
-n
(2.15)
Naturally, the performance of
the true
e;
e
if
similarly, while for
<
e'PT
then
.9'
e'PT
Q, -n
close to
2
n,a.
a
depends on
-n
is away from
L
e'PT
-n
(0 <
a <
1)
e
and
-n
as well as
may behave very
may behave better than
e .
-n
Following James and Stein (1961), we may consider the estimator
e'JS
_n
defined by
e'JS
(2.16)
where
-n
Ln
= {1
- aiL
is defined by (2.13) and
}e,
•
n -n
a
is an appropriate constant,
,
which we may discuss later on.
Though the existence and convergence of
moments of R-estimates of location have been studied by Sen (1980b),
there seems to be a basic problem with the study of the risk of
according to (1.3); this is mainly due to the fact that
though it could be very small. and as such,
moment of any positive order.
L-n l
'JS
2n
p~{Ln = O} >
o.
. may not have a finite
To avoid this technical problem and to
introduce more flexibility in the definition of the shrinkage estimator,
we consider the following modifications.
Parallel to (2.4)-(2.5), we let
$. (kl (n+l» •
)
for
k=l, ... ,n; 1·:$; j
1 n
(2.17)
m
'0
n,J~
~
P,
a . (k) = E$. (U k)
. nJ
) n
and define
= n- Li=l a nJ.(R..
)a .(R.J.
~J
n~
~~
M
......n
or
= «m
j.£=l ..... p,
. 0»
n/J~
by
•
7.
•
•
where
R
= rank of
ij
Then
X
ij
M
among
x1j, ... ,X
1 S i
for
,
$
nand
is a translation-invariant, robust and consistent
-n
defined by (2.6).
estimator of
nj
Also, we define
by letting
(2.18 )
where
= n
a
J.,{ T . <8 . nJ nJ
A
is some pre-fixed positive number.
d
Finally, let
•
c: 0 <
n
; ch (Qf)
p
,..,...0
< 2(p-2)
C
be the smallest characteristic root of
Qf
.-...0
be a given number; naturally, we assume that
p
~
3.
Then our proposed estimator is
0,
(2.20)
where
8 ; {
-n
E(> 0)
e
and
-n
tilrough
£
;
£
n,a
is an arbitrarily small number,
by (2. 3) .
Ln
(being
3.
(I.
<
or
" d,
(which would require
(3.1)
o ,
-n
L
< £
,
if
Ln
~ E
,
n
L
n
is defined by (2.13)
though we are not adapting (necessarily)
to be close to
a
,,£
L
n
>
1) •
Also, the James-
O.
Preliminaries on asymptotic admissibility.
3), for any
if
Note that the idea of PTE is incorporated in (2.20)
Stein rule is adapted here for
•
Let then
r
_n
(2.19)
and
a/.In) - T . (8 . + a/.In)}/ (2a), 1 ,; j ,; p,
nJ nJ
Note that according to
if there exists some other estimator
p
(0,8),
n - -
0*,
-n
such tha t
8.
~,
with strict inequality holding for some
2.
inadmissible estimator of
.
hm
(3.2)
n
~
p
~
(0 * ,e) "
n -
-
then
a
•
is termed an
-n
If instead of (3.1), we have
(a,e),
lim p
n -
n~~
with strict inequality holding for some
V ~.'
.~,
then
a
_n
is termed
asymptotically inadmissible (for fixed alternatives).
For the mean of
a p-variate normal distribution, it has been observed by Judge and Bock
e
(1981', p.172) that for values of
at' or close to the origin, the
risk advantage of the James-Stein estimator, relative to that of the
maximum likelihood estimator, could be considerable.
To exploit this
result fully in the context of nonparametric estimation, we conceive of
{K}
n
a sequence
Kn.....
'Xl' ... 'X
""'Il
d.f.
replace
(more precisely,
- n
F{~
e
-~
l),
by
n -~ ).
~
~
p
lEE.
are LLd. r.v. with the
Then, in (1.2)-{l.4), we
and denote the corresponding risk by
* (a_* _
pn
,).).
n (0*
__
lim p*
,).) " lim
n
X l'''.'X
)
......nn
......n
for some given
similar notation is used for
(3.3)
•
of local translation alternatives, where under
n
~
p * (0 * ,).).
n -
A
If
n __
p*{a,).),v).
E EP ,
~
with the strict inequality holding for some
A,
then
a
_n
is termed
asymptotically inadmissible for local translation alternatives.
The main objectives of this study are two-fold: (i) To establish.
the asymptotic admissibility of
to (3.2», and
for fixed alternatives (according
(ii) to establish the asymptotic inadmissibility of
for local translation alternatives, according to (3.3).
for the sake of compatibility, we confine
ourse~ves
e
_n
In either case,
to the class of
•
9.
•
R-estimators.
Note that for (3.3), we have to deal with a triangular
array of (row wise) i.i.d.r.v.
(whose d.f. may depend on
may cause some problems with manipulations.
However,
n), and this
a
is a translation-
-n
invariant estimator, and hence, most of these problems can be avoided by
X.
.....nJ:
replacing
location
4.
0
X . - n
by
-J.,
-nl.
A, 1 S i S n,
which are i.i.d.r.v. with
K ).
(under
n
Asymptotic admissibility results.
We shall consider first the fixed
alternative asymptotic admissibility results.
a ,
_n
some moment convergence results on
•
In this context, we need
and these we introduce first.
We assume that there exist a positive number
a
(not necessarily greater
than or equal to 1), such that
(4.1)
It follows from Theorem 2.1 of Sen (19BOb) that for monotone
V
jj
> 0,
for every
k(> 0),
there exists a sample size
(4.2)
lim sup E
F
n +
~
Ianj
- e. I
k
< m, V k: 0 < k < m.
)
We need, however, results stronger than (4.2).
additional regularity conditions.
.
J(~l, ...
•
where
0«
J.,),
nO(~nO(k,a»,
and further
such that
each
with
,p),.•.(r) (u)
(0 < u < 1),
such that
)
~
Hence, we introduce some
As in Sen (19BOb), we assume that for
(dr /du r) •. (u), r = 0,1,2,
)
and there exist positive constants
exist almost every-
KO
«~)
and
10.
•
(4.3)
As in (2.7), we assume the existence of
further that for some
n
f[j]
almost everywhere and
> 0,
(4.4)
Then', from Theorem 2.2
that for each
and (2.49)-{2.50)
j{=l, ..• ,p),
(4.5)
n(e.-e.) =
nJ
)
n-~Iw
where
of Sen (1980b), it follows
.1
+ 0
nJ
-1
~.
)
'
nT .(e.) + w ' ,
nJ
nJ )
almost surely (a.s.) as
o
(4.6)
n
+~,
0
<~,
so that
In passing, we may remark that (4.3) holds for the Wilcoxon
scores
other
•
,V k < (1 - 20)/0 = ,0 * , say.
In the sequel, we assume that (4.3) holds for some
0* > 2.
and
Normal scores (0
(0=0),
~ornmonly
(for some
__
lim p n (e,e)
(4.7)
.'
0) and all the
,
used score functions.
Note that by (1.2),
(4.3) and (4.4)
arbitrarily close to
n +
00
(1.3),
0 <
(2.3),
(4.5) 'and (4.6), under (4.1),
~),
Tr(Q lim nE(~n -~) (~n -, ~).)
n+
= Tr (Qrl
OO
,
..J'"
where
r
is defined by (2.6)-(2.10) and under (4.3),
Tr(Qrl < "'.
"'"
we note that by (2.15),
. .
(4.8)
n(e'Qe )I{L < i
),
....n,:.,.::r...n
n
n,a
Next,
•
'.
11 •
•
so that by the Holder inequality, we obtain that
l-.!.
< i
(4.9)
where
q(> 1)
is some positive number.
n,Cl }]
q
,
Note that by (4.2), under (4.1),
(4.10)
~
• n " n O·
On the other hand, by (2.13),
R. D/a } S
•
2
"max (nT ./m* ..
n lSjSp
n) n,1)
L
min
lSjSp
2 < n -1 R.
PT.
{
oJ
(4.11 )
=
min p{IT
lSjSp
.1
< n-"'(i
n)
m* ..
n/a n,]]
J,
so that
*}
m
..
n/n n,))
J"'}
Now, incorporating' Theorem 3 of Sen (1970) into the proof of his Theorem 1,
we conclude that under (4.3) with
·1
p{ ITn)
ej
whenever
converges to
j=l, ... ,p).
~
0
(as
n ~ m)
whenever
~
+2
e.)
(i.e.,
0
+-
-n
(in a local sense).
may not have a finite risk even asymptotically as
proceed to study the case of
•
for some
e
= 0
or close
After (2.16), ,we have noticed that because
L
of the positive probability mass (however small) of
(4.12)
+0
and (4.4), for every
Hence, under (4.1), (4.3), (6 < ~)
ApT
are asymptotically risk equivalent. We
0, e
and e
shall see later on that this feature does not hold for,
to
+0,
m* .. )"'Ie.} = 0(n- ), as n ~~. Thus, if in (4.9) ,
D,n O,]J
]
q > 2, then by (4.10) and (4.11), we conclude that (4.9)
2
S n-"'lR.
we choose
(fixed)
<~,
6
"S
e
_n
in (2.20).
at
n
n
+
Note that
m.
0,
"JS
e
_n
Hence, we
12.
•
Now, proceeding as in (4.9) through (4.11), it follows that
(4.13)
E{I(L n
lim sup
n -+ 00
Al so, note tha t
d
(Qr ),
n = chp_n
so that
2"'-1 -1'-1'
n~n-n
d e r.........
Q rD_O
e
(4.14)
(n~~n) {d~ r~~i:12-li:l~/i~~n]}
=
~ (nS'QS )d2chl(r-lQ-lr-lQ-l)
_~n
n
Therefore choosing some
(4.15)
I
(L n
~
-n.....
E'
(>
-n .....
0)
arbitrarily small, we have
2 2 -2 " ' - I -1'-1'
E)C d L ne r Q r e
n n . . . n.::.n..... .....n_D
,;; I(L n
Ln
= I(E ,;;
,;; I(L n
•
2 -2 "
'
ne Qe
n .....n---n
dc L
~
< nE')c
2
2 -2 " '
Ln ne. . . .n.;;;J....n
Qe +
2
< nE')c E- ns'Qs
......n.. . . . . .n
+
I(L n . ~
nE')c
2 -2 " '
Ln ne.......n.:.::..
Qe
I(L n ~ n£')c1£~2{l
n
<
S'QS } •
. . . .n;..;......n
Now proceeding as in (4.9)-(4.11) , the first term on the right hand side
of (4.15) converges in the first mean to
n-+ 00) ,
0 (as
when
+0,
!
for the second term we make use of (4.2) and conclude (by using the Holder
inequality) that this also converges in the first moment to
for every fixed
under (4.1),
(4.3)
e.
(0
Hence, we obtain that for every (fixed)
<
'<)
and (4.4), as
0 (as
e
n -+ (0)
+0,
n -+ 00,
(4.16)
so that for every (fixed)
e
4 .0,
~
and
e
_n
are asymptotically risk
•
13 •
•
equivalent.
Again, we shall see later on that the situation is different
for local translation alternatives.
We proceed on next to study the asymptotic inadmissibility of
en
the classical R-estimator
considered before (3.3).
{X
ni
n,
e
=
e
-n
= n
(4.17)
•
We conceive of a triangular array
, 1,;; i,;; n; n ~ l}
where
-"> A,
of row-wise LLd.r.v. with the d.L
K
and, as before, let
for local translation alternatives, as
{F(n)}'
stand for the hypothesis that for the given
n
i. e. ,
F( )
n
A is finite
.
= F (x - n
(x)
(€
"'"
"'"
Ff?)
-"> A),
"'"
x
"'"
€
-P
t.-
,
and fixed, and the d.f. F
satisfies all the
regularity conditions stated in earlier sections.
For the PTE
'PT
e
-n
in (2.15), we have, under
K
n
in (4.17),
(4.18)
= (A'QA)I(L
"'" __
~1 n,a )[n(e_n - _0
e ) • Q(e n
n < 1n,a ) + I(L n
- _n
e )] '
so that by (2.14) and (4.17),
(4.19)
•
We denote by
H (';6)
p
the non-central chi square d.f. with
p
degrees
14.
of freedom (DF) and non-centrality parameter
~(~
0).
Then, we have
•
[see Sen and puri (1967)]
(4.20)
P{Ln
lim
n ... "
S
~(a) IKn }
2
-1 )
= H ( X (a);A'r A
p
p
•
..............
For the second term on the right hand side of (4.19), we make use of
(i) the incomplete moments of multinorrnal distributions [c.f. Section 4
of Sen and Saleh (1979)
of the
IT
nj
(e
lim
(4.21)
n--
4
j
)1
J.
(ii)
(2.11). (4.5), (4.6) and
(under (4.3) for
E{r (L n ~
- 2H
0<
t),
(iii) the integrability
and obtain that
/(a»)n(e
- ......n_
e ) 'Q(e. . . n - .......o
e ) IK n }
p
..... n
p+
2 ( X2 (a)iA • r -l~
A + H +4 ( X2 (a);A ' r -l~]
A •
P
-P
P
--
•
I
I
!
From (4.19) through (4.21), we have
(4.22)
. • -1 A)]
+ (A • QA) [ 2H 2 ( X2 (a); A• r -1 A) - H +4 (
X2
(a); A r
-p+
P
-- P
P
---
We may note that (4.7) holds as well for
that for the classical R-estimator
* ,
~,
in (4.17), so
we have
__
lim p n (e,
A) = Tr (Qr) ,v A
(4.23)
{K n }
€
Ep •
n--
Since
H (x;o) ;, H
q
.(x;o),
q+1
vi
~
of (4.22) is bounded from below by
1, x;, 0,
a
~
0,
the right hand side
•
15 .
•
.
(4.24)
Thus, for all
,
~ ~ ~
A for which
Tr(Qrl,
the difference may not be large for any
A
, -1
r
A
~
+
On the other hand, for all
~.
(4.24) exceeds (4.23), though
A and it converges to
A:A •QA"S
~Tr(Qr),
0
as
the right
hand side of (4.22) is bounded from above by
(4.25)
we that the asymptotic risk of the PTE is smaller than that of
,
A:A QA S
- --
~Tr(Qr).
-
For
~:~Tr(Qr) <
•
A QA < Tr(Qr) ,
for
towards the lower
boundary (4.23) is greater than (4.22) and conversely towards the upper
•
boundary, though the actual difference may be quite small.
-PT
e
_n
Thus, the PTE
turns out to be a robust competitor, with better performance for
close to
0
and not so poor for
A away from
O.
However, in
accordance with (3.3), we can not conclude that either
e
_n
or
The picture is different with respect to
By
(2.20), we
have
-s
nee
......n
'-S
- e ) Q(e
= I
_0
(L
...., .... 0
- e )
......0
,
n
< d (A QA) + I
(L
- _n
e)
n
2 2" -2 -'--I -1"-1" }
+ c d L ne r Q r e .
n n . . . n-n...... -n ......n
Now, by (4.20), as n
•
(4.27)
~ w,
I
is
A E ~.
inadmissible with respect to the other for all
(4.26)
A
•
A QAH (EIA r
................ p
..... -
-1
A)
16.
•
while proceeding as in (4.21)-(4.22),
(4.28)
lim E{I(L '
n
n-
=
~
E)n(e
....n
e ) 'n(8 -' -n
e ) IKn }
-
.....0 ) 5 -n
{I - Hp+2(q~'r-l~)}Tr(2!:)
I
- 2H
p+
"-1
2(E;A r
...........
A)
,-
(~'~){Hp(E;~'2:-l~)
+ H 4(E;A , r -1 A) } •
p+
1'0#
- -
-
Also we may note that
(4.29)
d
2 ,.'-1 -1'-1'
ne r 0 r e
n .....n......o ~ .....n-n
•
(4.30 )
e )'r-l(l)
- e')
,,{Cn(e - .....n
.....n ......o
......n
-n
"{n(~ - e
.....n
)'Q(e
....
-
e ) •
......n.....n
• d ][dne'r-le
n
ne'Qa
. . . n..........n
n .....n......n .....n
}J,
Notice that because of the translation invariance of '
hold under
every
(n(2
n
{K }
n
as well (with
k (> 0), under
- 2n)
K ,
'2(~n ~ ~n»k
n
with
e
n
J}J,
peing replaced by
e
(4.5)-(4.6)
-n'
e ),
_n
so that for
adequately large,
are integrable.
Further, making use of the
Juretkova-linearity of rank statistics in the multivariate case [viz.,
Sen and puri (1977)], it follows that under
{K },
n
as
n
~
00,
•
,
17.
•
\'
,
.... -1'"
= ne-nr .e. . . 0
L
(4.31)
n
+ n
n
where
o ,
(4.32)
as
0,
Therefore, on the set
',-1')
r e
Ln-1 - (ne-n....
....0
(4.33)
_P o ,
n
as
0-+
00
•
by (4.5)-(4.6),
{K},
Finally, under
-1
{K },
under
n
(4.34)
•
Let us now denote by
~ =
with mean vector
w
a p-vector having the multinorrnal distribution
r-'2 ~
and dispersion matrix
I
-P
.
Also, let
be a random variable having the noncentral chi square d.f.
H(":lI).
q
Then, by using (4.29)-(4.34) along with the desired integrability
conditions, we conclude that for every
(4.35)
lim
n--
=
where
(4" 36)
E{I(Ln ~
E)d
L- l n(6. . . .0
n n
Chp(QrJ{E(X~2)
l
II = ).'r- ).,
-- -
.
e ) 'r-le
.......0 .......0
.......0
E[I(Ie'~
-
0,
E >
IK}
n
< E) (!e'!j)-lW'r-'2lJ}
and
lim E{ I (L
n
n--
~
E)d 2 L-2 ne"'-1
r Q-1'-1'
r e
nn
.....n....n ............n......n
l
IK }
n
l
)
= [ch (QrJ]2{Tr(Q- r- )E(X-\ lI) + II , E (-4
Xp +4 ,lI
p-p+,
•
, -2'
I(!e' ~ < E) (Ie~)
Ie~) }
- E(
where
lI' = ).'r-1Q-lr,""1).
-- - - -
and
A =
-I, -1 -'2
- -Q
r
r
.
Now
18.
•
(4.37)
= 1I
~
e
E
-~1I
,;
1I~ e-~lI
,;
[lp/2
Jo
fr=O .!.,
(~lIlr[ IP/2 2P /2)-1
r.
x
r+(p-3l/2
..
e
-~x
dx
Er +(P-ll/¥<'r+(p_ll/2l
2P/2)-11l~E(P-ll/2e-~lI(l-E),v P"
2,0 < E:!Il.
Also,
(4.38 )
•
•
• -2' ]
E [ l(w W < El (w W) w AW
_.....
.... ..........
-
,; E [1 (w'W < E) (w 'Wl -lTr (A) ]
..... ....
..... .....
.....
e
Therefore, whenever
(4.39)
p" 3,
-~1I
(1-El
,VP"2,O<E';1.
from (4.26) through (4.38), we obtain that
lim p *('5
8,),)
n-><o n = Tr(Qrl{l - H + (Ellll} + (~'~l [2H + (Ellll - Hp+4(Ellll]
p 2
p 2
2
2
- 2c ch (Qr)E(x-:
) + c [ch (Qrl]2{TrlQ-lr-llE(x-42 )
p -p+2,1I
P -p+ ,lI
J
c 2 [ ch (Qf) 2 E{l(w I W) <
P ........
£)
(w I W) -2'
w AW),
•
19.
•
,
where the last term is non-negative, while the term before last is
bounded from above by
2c Ch (E!:lE(P- l l/2/(2 P/2/p /21,
p
Hence, we have for every
-
(4.401
lim
P: (2s..~1
E: 0 < E S 1, A
€
uniformly in
},.
~,
S Tr(.2.!:1 [1 - Hp+2(E:61] +
(~'~l
[2Hp +2 (E:6l
2 2
2
l l
- H + (E:61] - 2c'ch (QrlE(x.) + c Ch (Qrl [Tr(Q- r- lE(X- 4 .)
p
p+,u
p _ _
p+2,u
p 4
2
+ 6*E(X~4)J + 0(E(P'-11/2)
=
•
{Tr (.2.!:1
2
2
2
= {Tr(Qrl - 2c'ch (QrlE(x.) + c ch (Qrl
2
p p+,u
p -
+
6*E(X~4'6)]}
+ 0(E(p-l)/2) +
l l
4
)
[Tr(Qr- lE(X,..,; _
p+2,11
o (E P/ 2),
uniformly in
A.
Now, by the results of Section. 2 of Sclove, Morris and Radhakrishnan (1972),
the leading term on the right hand side of (4.40) is
YC: 0 < C < 2(p-2l,
and
p
~
3, _n
e
Hence,
* •
E(> 0)
chosen sufficiently small and for
translation alternatives.
•
0 < c < 2(p-2l,
is asymptotically inadmissable with respect to
also exhibits the asymptotic inadmissibility of
:
Tr(.2.!:) ,
__
=limp n (e,A),
(4.411
so that for
A. EP •
<
'5
e .
_n
for local
This
20.
5.
Some General Comments.
As has been mentioned after (2.16) for the
•
James-Stein type estimator in (2.16), one faces a technical problem in
eval~ating
Ln = 0,
the risk, as
with a positive probability.
proposed estimator in (2.20) eliminates this problem.
The
It may be noted
that rank statistics are not invariant under linear transformation, hence
Q and
general loss function with matrix
this context, the choice of
out.
E(> 0)
and
r
is considered.
c(> 0)
However, in
remains to be worked
Because of (4.37)-(4.38), we need to restrict
E: 0 < E S 1,
in (4.40),to derive the inequality in (4.41) we need to restrict
o
< c < 2(p-2).
restricted to
In the simple James-Stein (1961) estimator,
c = (p-2).
c
c
to
has been
As a general rule, the same choice of
be recommended in (2.20).
while
c
may
However, for the normal mean case, the study
of Berger, Bock, Brown, Cassella and GIeser (1977) shows that for small
values of
n, c(S c
n,p
)
has a range depending on
may be quite small compared to
practical point of view, for
use a smaller value of
c.
p-2
n
when
n
nand
is small.
p
and
£
n,p
Hence, from a
not very large, we would be tempted to
Ideally,
E(> 0)
should be as small as
possible, so that in (4.40), these terms are really negligible.
as
c
•
However,
becomes small, (4.35) and (4.36) demand more larger sample sizes
for a valid justification.
Hence, the choice of
made with relation to the sample size.
E 'also needs to be
Some numerical studies
rray be more appropriate to study the elegancy of the asymptotic theory
for moderate sample sizes.
translation alternatives,
respect to
Finally, we may remark that though for local
e
_n
is asymptotically inadmissible with
is not so,;
Actually, for
close to
the
•
.
21.
•
PTE performs better than
e·s ,
-n
though for some
may perform better than
shrinkage estimator
"
•
•
e·s
-n
-'PT
e •
-n
does not dominate over
for all
Hence, the proposed
e
-n
everywhere.
22.
•
REFERENCES
1. Berger, J .0. (1980). A robust generalized Bayes estimation and confidence
region for a mUltivariate normal mean. Ann. Statist. 8, 716-761.
2. Berger, J.O., Bock, M.E., Brown, L.D., Castella, G. and GIeser, L. (1977)
Minimax estimation of a normal mean vector for arbitrary quadratic loss
and unknown covariance matrix.
Ann. Statist. 5, 763-771.
3. James, W. and Stein, C. (1961). Estimation with quadratic loss.
Fourth Berkeley symp. Math. Statist. Prob. 1, 361-379, Univ. of
Proc.
California Press.
4.Juuge, G.G. and Bock, M.E.
(1978).
The Statistical Implications of Pre:T~e~s~t~a~n~d~S~t~e~~~'n~-~R~u~l~e~E~s~t~i~m~a~t~o~r~s=-~i~n~E~c~o~n~o~m~e~t~r~~~·c~s. North-Holland Publishing
Company.
Amsterdam.
5. puri, M.L. and Sen, P.K. (1971). Nonparametric Methods in Multivariate
John Wiley & Sons, Inc. N.Y.
Analysis.
6. Saleh, A.K.Md. Ehsanes and Sen, P.K. (1978).
llonparametric Estimation
of location parameter after a preliminary test on regression.
Ann.
Statist. ~, 154-168.
7. Saleh, A.K.Md. Ehsanes and Sen, P.K.
(1983).
Least Squares and Rank
Order Preliminary Test Estimation in General Multivariate Linear Models.
Proceedings of the Golden Jubilee of the Indian Statistical Institute.
•
To appear.
8. Sclove, S.L., Morris, C., and Radhakrishnan, R.
(1972).
Non-optimality
of preliminary-test estimators for the mean of a multivariate normal
distribution.
Ann. Math. Statist.
il,
1481-1490.
9. Sen, P.K.
(1970). On some convergence properties of one-sample rank
Statistics. Ann. Math. Statist. 41, 2140-2143.
10. Sen, P.K.
(1980a).
order statistics.
On almost sure linearity Theorems for signed rank
Ann. Statist. ~, 313-321.
11. Sen, P.K. (1980b). On nonparametric sequential point estimation of
location based on general rank order statistics.
Sankhya, Series A, 42.
201-218.
12. Sen, P.K. and Puri, M.L. (1967). On the theory of rank order tests for
location in the multivariate one-sample problem. Ann. Math. Statist. ~,
1216-1228.
13. Sen, P.K., and Puri, M.L.
(1969). On robust nonparametric estimation in
some multivariate linear models.
Multivariate Analysis II (edited by
P.R. Krishnaiah) Academic Press, N.Y. 33-52.
•
G
.
23.
'/
•
14. Sen, P.K. and Saleh, A.K.Md. Ehsapes (1979).
Nonparametric estimation
of location parameters after a preliminary test on regression in the
multivariate case. Jour. Multi. Analysis, ~, 322-331.
15. Stein, C. (1956).
Inadmissibility of the usual estimator for the mean
of a multivariate normal distribution.
Proe. Third Berkeley Symp. 12th.
Univ. of California Press •
Statist. Prob. !' 197-206.
•
•
•