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Whither jackknifing in stein-rule estimationPranab Kumar Sen aa Department of Biostatistics , 20Th University of North Carolina , Chapel Hill,NC, 27514Published online: 02 Nov 2010.
To cite this article: Pranab Kumar Sen (1986) Whither jackknifing in stein-rule estimation, Communications inStatistics - Theory and Methods, 15:7, 2245-2266, DOI: 10.1080/03610928608829246
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COMMUN. STATIST.-THEOR. METH., 1 5 ( 7 ) , 2 2 4 5 - 2 2 6 6 ( 1 9 8 6 )
NHITHEE JACKKIdIFIMG I N STEIN-RULE ESTIMATION ?
Pranab Kumar Sen
Department o f B i o s t a t i s t i c s 201H U n i v e r s i t y o f l i o r t h C a r o l i n a
Chapel H i l l , i l C 27514
Key Words and Phrases : Asymptotic risk ; bias ; bias-reduction ; inahissibiZity ; jackknifing ; jackknifed estimator of dispersion matrices ; minimaxity ; Pitman-alternatives ; risk-estimation ; shrinkage estimators.
ABSTRACT
I n mu1 t i -pa ramete r ( mu1 t i v a r i a t e ) e s t i m a t i o n , t h e S t e i n r u l e
p rov ides minimax and admiss ib le e s t i m a t o r s , compromising g e n e r a l l y
on t h e i r unbiasedness. On the o t h e r hand, t h e p r i m a r y aim o f j a c k -
k n i f i n g i s t o reduce t h e b i a s o f an e s t i m a t o r ( w i t h o u t n e c e s s a r i l y
compromising on i t s e f f i c a c y ) , and, a t t h e same t ime, j a c k k n i f i n g
p rov ides an e s t i m a t o r o f t h e sampling v a r i a n c e o f t h e e s t i m a t o r as
w e l l . I n shr inkage e s t i m a t i o n ( where m i n i m i z a t i o n o f a s u i t a b l y
d e f i n e d r i s k f u n c t i o n i s t h e b a s i c goal ) , one may wonder how f a r
t h e b i a s - r e d u c t i o n o b j e c t i v e o f j a c k k n i f i n g i n c o r p o r a t e s t h e dual
o b j e c t i v e o f n i i n imax i t y ( o r a d m i s s i b i l i t y ) and e s t i m a t i n g t h e
r i s k o f t h e e s t i m a t o r ? A c r i t i c a l a p p r a i s a l o f t h i s b a s i c r o l e o f
j a c k k n i f i n g i n shr inkage e s t i m a t i o n i s nade here. R e s t r i c t e d , semi-
r e s t r i c t e d and t h e usual ve rs ions o f j a c k k n i f e d sh r inkage es t imates
a re considered and t h e i r performance c h a r a c t e r i s t i c s a r e s t u d i e d . It i s shown t h a t f o r Pitnian-type ( l o c a l ) a l t e r n a t i v e s , u s u a l l y ,
j a c k k n i f i n g f a i l s t o p r o v i d e a c o n s i s t e n t e s t i m a t o r o f t h e ( asymp-
t o t i c ) r i s k o f t h e shr inkage e s t i m a t o r , and a degenerate asympto-
t i c s i t u a t i o n a r i s e s f o r t h e usual f i x e d a l t e r n a t i v e case,
Copyright O 1986 by Marcel Dekker, Inc
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SEN
1. INTRODUCTION
L e t X_, ,, .. ,X,, be n (> - 1 ) independent and i d e n t i c a l l y d i s t r i -
buted random v e c t o r s (i. i.d.r.v.) w i t h a p ( - > 1 ) - v a r i a t e d i s t r i b u -
t i o n f u n c t i o n Cd.f.1 F hav ing a f i n i t e mean v e c t o r -" 0 and a p o s i t i v e
d e f i n i t e (p.d.) d i s p e r s i o n m a t r i x X , where b o t h 0 and C a r e unkn- .d -" .d
own. When F i s assumed t o be mu l t i -no rma l , t h e maximum 1 i k e l i hood
e s t i m a t o r (m.1 .e.) o f i s t h e sample mean v e c t o r Tn = nn1yn i = l - i X a
and t h i s possesses some op t ima l p r o p e r t i e s too, However, f o r p 3,
S t e i n (1956) showed t h a t Tn i s n o t admissible, and es t in ia to rs o f e -"
dominat ing over Tn , i n r i s k , known as t h e Stein-ruZe estimators , have been considered by James and S t e i n (1961) and a hos t o f o t h e r
workers. A v e r y d e t a i l e d account o f t h e S t e i n - r u l e e s t i m a t i o n t h e -
o r y f o r mu1 t i - n o r m a l F i s g iven i n Berger (1980) as w e l l as i n some
o t h e r contemporary tex tbooks on m u l t i v a r i a t e s t a t i s t i c a l a n a l y s i s .
Though such S t e i n - r u l e e s t i m a t o r s have been s t u d i e d f o r some
s p e c i f i c non-normal F, i n genera l , f o r an a r b i t r a r y F a t h e c o n s t r -
u c t i o n o f such S t e i n - r u l e e s t i m a t o r s and v e r f i c a t i o n o f t h e i r ( ex-
a c t ) dominance o v e r t h e co r respond ing m . 1 . e . ' ~ a r e y e t t o be made
i n f u l l g e n e r a l i t y . Considerable d i f f i c u l t i e s a r i s e i n t h e s tudy
o f t h e (exac t ) d i s t r i b u t i o n t h e o r y o f such e s t i m a t o r s ( f o r f i n i t e
n ), and these, i n t u r n , make i t d i f f i c u l t t o e s t a b l i s h t h e a n t i c i -
pated dominance o f t h e S t e i n - r u l e es t imato rs . On t h e t o p o f t h a t , f o r normal as w e l l as non-normal d i s t r i b u t i o n s , t h i s dominance o f
t h e S t e i n - r u l e e s t i m a t o r ho lds o n l y i n a s h r i n k i n g neighbourhood
o f t h e p i v o t ( i n t h e sense t h a t as t h e sample s i z e n increases,
f o r any f i x e d d e v i a t i o n o f t h e t r u e parameter p o i n t f rom t h e p i v o t ,
t h e r a t i o o f t h e r i s k s of t h e S t e i n - r u l e e s t i m a t o r and t h e c o r r e s -
ponding m.1.e. converges t o one, and t h e a n t i c i p a t e d dominance may
h o l d o n l y when t h e t r u e parameter 0 l i e s i n a b a l l w i t h t h e p i v o t
as t h e c e n t r e and hav ing a r a d i u s " o f t h e o r d e r n-' 1. Never the less,
f o r F n o t n e c e s s a r i l y mu1 t i -no rma l and f o r genera l estimable parum-
eters , under a p p r o p r i a t e moment-condit ions, t h e asympto t i c dominance
o f a S t e i n - r u l e e s t i m a t o r has been s t u d i e d i n a more genera l setup
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION? 2247
C of U-s t a t i s t i c s 1 by Sen (19841, where the usual Pitman-type (i.
e. , l o c a l ) a l t e r n a t i v e s C t o the p ivo t ) have been adapted t o pro-
vide a meaningful and c l e a r p ic ture of t h i s asymptotic dominance.
A very s imi la r p ic ture holds f o r the Ste in-rule est imators r e l a t ed to the m.1 . e O 1 s f o r an a r b i t r a r y F I v iz . , Sen(1906 )I as well as f o r the usual l i n e a r est imators i n l i n e a r models 1 v i z , , Saleh and
Sen (l96G)l.
The theory of Ste in-rule est imation has a l so been considered i n some nonparametric setups ( r e l a t ing t o possible non-linear e s t - imators ) by Sen and Saleh (1985a,b) and Saleh and Sen (1985). The
essent ia l c h a r a c t e r i s t i c s of the Stein-rule est imation theory a r e i n t a c t i n t h i s asymptotic nonparametric setup too, In nonparamet-
r i c inference , jackknifing plays an important ro l e , Often, t o re- duce the bias of an est imator o r t o est imate i t s sampling variance, jackknifing i s very successful ly employed. This brings us t o the natural question : Mhat i s the ro le of jackknifing in the Ste in-
ru l e est imation theory ? Our primary object ive i s t o address t h i s
issue.
A S te in-rule est imator i s genera l ly biased. This i s mainly due
t o the f a c t t h a t in the shrinkage f a c t o r , one incorporates a t e s t s t a t i s t i c (on the adequacy of the p ivot) which introduces non-lin-
e a r i t y , and t h i s in turn introduces bias. In a non-normal o r non- parametric setup, of ten , t h i s bias i s too complicated t o be evalu- ated i n an exact form . For an a r b i t r a r y ( and,possibly, biased )
es t imator , jackknifing not only reduces the order of magnitude of the b ias , but a l so provides an est imator of the sampling variance of the est imator ( o r the corresponding r i s k when i t i s convenient- l y defined i n t e rns of the sampling dispersion matrix of the estim- a t o r 1. An inherent reversed martingale s tmcture of jackknifing C explored sys temat ica l ly by Sen (1977)) provides a c l e a r p ic ture of the mechanism of t h i s dual ro l e of bias-reduction and estimation
of the r i sk . In the t r ad i t iona l case ( without having any pivot in
mind), f o r the mul t iva r i a t e location problem, the r i s k of the m.1.e.
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i s t r a n s i a t i o n - i n v a r i a n t , and hence, i s a c o n s t a n t [depending on
t h e o t h e r nu isance parameters a s s o c i a t e d w i t h t h e u n d e r l y i n g d. f , ) .
T h i s cons tan t r i s k can be v e r y e f f e c t i v e l y e s t i m a t e d by t h e usua l
j a c k k n i f i n g technique, However, t h e p i c t u r e may be q u i t e d i f f e r e n t
i n t h e case of sh r inkage e s t i m a t i o n , as w i l l be d e a l t w i t h here.
F o r any f i x e d a1 t e r n a t i v e (i .e., t h e t r u e parameter be ing d i f f e r -
e n t f rom t h e p i v o t ) , a s y m p t o t i c a l l y , a S t e i n - r u l e e s t i m a t o r and i t s
j a c k k n i f e d v e r s i o n a r e b o t h e q u i v a l e n t , and n e i t h e r dominates over
t h e conven t iona l one. I n v iew o f t h e i n e f f e c t i v e n e s s o f t h e S t e i n -
r u l e e s t i m a t i o n t h e o r y f o r f i x e d - d l t e r n a t i v e asympto t i cs , as has
been nlent ioned e a r l i e r , t h i s o v e r a l l equ iva lence o f j a c k k n i f e d and
t h e usual ve rs ions o f t h e S t e i n - r u l e e s t i m a t o r s i s o f no p r a c t i c a l
s i g n i f i c a n c e . however, f o r P i tman-type ( l o c a l ) a1 t e r n a t i v e s , t h e
s i t u a t i o n i s q u i t e d i f f e r e n t . I n such a case, t h e conven t iona l es-
t i n a t o r Tn , t h e r e l a t e d S t e i n - r u l e e s t i m a t o r and i t s p l a u s i b l e Ja-
c k k n i f e d v e r s i o n s a r e g e n e r a l l y n o t a s y m p t o t i c a l l y e q u i v a l e n t , and
t h e i r r e l a t i v e r i s k p i c t u r e s c l e a r l y convey some meaningfu l asymp-
t o t i c dominance r e l a t i o n s , and these w i l l be s t u d i e d here.
For s i m p l i c i t y o f p r e s e n t a t i o n s , we c o n s i d e r here t h e case o f
t h e m u l t i v a r i a t e l o c a t i o n problem a l t h o u g h a s i m i l a r t h e o r y works
o u t w e l l f o r genera l e s t i m a b l e parameters. S ince t h e S t e i n - r u l e es-
t i m a t o r , g e n e r a l l y , i n v o l v e s a t e s t s t a t i s t i c ( f o r t h e a p p r o p r i a -
teness o f t h e p i v o t ) i n a d d i t i o n t o t h e sample mean v e c t o r and co-
va r iance m a t r i x , i n t h i s setup, j a c k k n i f i n g may o n l y be a p p l i e d t o
t h e mean v e c t o r , o r t o t h e t e s t s t a t i s t i c and mean vec to r , o r t o
a l l t h e t h r e e s e t s o f random elements. These r e l a t e t o t h e restric-
!., I, ::c'rni-r~cst-rictecl and unrestricted v e r s i o n s o f j a c k k n i f e d s h r i n -
kage es t imato rs . These v e r s i o n s a r e a l l f o r m a l l y i n t r o d u c e d i n Sec-
t i o n 2, S e c t i o n 3 dea ls w i t h t h e i r asympto t i c r i s k p r o p e r t i e s f o r
f i x e d a l t e r n a t i v e s , and t h e genera l equ iva lence r e s u l t s a r e presen-
t e d the re . The main r e s u l t s on t h e i r asympto t i c r e l a t i v e r i s k p r o -
p e r t i e s f o r P i tman- type ( l o c a l ) a l t e r n a t i v e s a r e p resen ted i n Sec-
ti011 4 . it1 t h i s con tex t , t h e n o t i o n o f usymptotic 2istribv.tionaZ
r'i::ii i'A:)l?i i s i n c o r p o r a t e d i n t h e d e f i n i t i o n and e s t i m a t i o n o f t h e
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION? 2249
of asympto t i c r i s k , and t h e r o l e of j a c k k n i f i n g i n p r o v i d i n g con-
s i s t e n t e s t i m a t o r s o f t h i s asympto t i c r i s k i s c r i t i c a l l y examined.
I t i s shown t h a t i n t h i s asympto t i c setup, g e n e r a l l y , j a c k k n i f i n g
f a i l s t o p r o v i d e a c o n s i s t e n t e s t i m a t o r o f t h e ADR of a S t e i n - r u l e
e s t i m a t o r of 0 . T h i s f i n d i n g a lone niay r a i s e t h e genera l i s s u e .. o f implement ing j a c k k n i f i n g i n S t e i n - r u l e e s t i m a t i o n , and t h e con-
c l u d i n g s e c t i o n i s devoted t o some genera l d i s c u s s i o n s ( i n t h i s
v e i n ) on t h e r o l e o f j a c k k n i f i n g i n sh r inkage e s t i m a t i o n .
2. JACKKNIFED STEIN-RULE ESTIEIATORS OF 0 .. For an e s t i n ~ a t o r Ln = I(;, ,'" ,En) o f 8 , c o n s i d e r a quadratic ..
Loss function
where Q i s some g i v e n p.d. m a t r i x . Genera l l y , Tn may be chosen as - .. a t r a n s l a t i o n - i n v a r i a n t e s t i m a t o r , and hence, t h e c h o i c e o f 4 may
be made independent l y o f any i n f o r m a t i o n one niay have on 0 ( such
as t h e p i v o t which w i l l be i n t r o d u c e d l a t e r on) . The r i sk ( i .e . ,
t h e expected l o s s ) o f 3 , i s t h e n g i v e n by
P,,( 1, 8 = E & ( 1,; 8 ) = Tr(QZn) , ( 2 . 2 ) .. where En = nEg{(Tn - !)(I, - 0 ) l i i s t h e mean-product moment m a t r i x
o f nk ( I n - 0 ) : I n a s t e i n - r u l e e s t i m a t i o n problem, we a r e g i v e n a
p i v o t f o , and shr inkage o f in i s made towards !o . Because o f
t h e t r a n s l a t i o n - i n v a r i a n c e o f t h e sample mean v e c t o r Tn , we may
take, w i t h o u t any l o s s o f g e n e r a l i t y , f o = 0. Then, f o l l o w i n g t h e - genera l p r e s c r i p t i o n of Berger e t a l . (1977) ( though n o t necessa-
r i l y assuming t h a t t h e u n d e r l y i n g d . f . F i s m u l t i - n o r m a l ) , we may
c o n s i d e r t h e f o l l o w i n g S t e i n - r u l e e s t i m a t o r :
where I i s t h e i d e n t i t y m a t r i x of o r d e r p , cn i s a p o s i t i v e number
( converg ing t o a l i m i t c : 0 < c < 2 ( p - 2 ) ) , p i s assumed t o be p o t 2 -1 -1-
l e s s than 3, Tn = n!:,SQ Xn) i s t h e t e s t s t a t i s t i c ( f o r t h e n u l l
hypo thes is Ho: Bo= 0 i.e., t h e adequacy o f t h e p i v o t ! , dn = cl; (QS ) , w p ..-n
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2250 SEN
i s t he smal les t c h a r a c t e r i s t i c root of QS and the sample covari- -1 n W-n ,
ance matrix Sn = n XiG1 (=Xi - F n ) tzi - Tn) i s an unbiased estim-
a t o r of 5 . For mu1 ti-normal F and s u i t a b l e I cn ) , the minimaxity " S of en has been es tabl ished i n Cerger e t a l . (1977). Irl our study,
we do not assume t h a t F i s necessar i ly multi-normal. Under sui ta t ; le " S moment-conditions on F , t he dominance of !n over Tn ( i n an asym-
p to t i c se tup) follows from the r e s u l t s of Sen(1984) on more gener- al i l - s t a t i s t i c s . Ele rewri te (2.3) a s
"S -2 -1 -1 = ( : - T n bn )Tn ; ;n = cndn9 :n a -n (2.4)
-2 and note t h a t the th ree s e t s of random elements (v iz , , I n , fin and
?n "S a r e a l l based on z1 ,..., E n and appear in el, . !$low, j a c k k n i f i n ~
m y be used on a pa r t of these randor!! elements C o r a l l ) . Keeping t h i s in mind, we nay consider the following versions of jackknifed Ste in-rule es t imators of 8 .
*
( I ) Restricted jackknifed estimator. Since Fn i s the most re- levant pa r t f o r the est imation of 8 , we nay incorporate jackknif- ... ing only on t h i s p a r t , and , thereby, obtain the following jackkni-
fed est imator corresponding t o the S te in - ru le es t imator in (2.4) :
;SRJ = n-l z v S R -n 1=1 'n,i , (2.5)
where ?R. = ^S "SR
~ n . 1 n !n - ( n - l ) ~ n - i j i ) , i = 1 ,..., n, and
@R - -2 nyn -; . + n - ~ ( i ) - Q - Tn !n)Tn-l(i) ; ?!n-l(i) = .mj 9
(2-6 )
f o r i = l , , . . , n . I t i s easy t o ve r i fy t h a t
eSRJ = 2 , f o r every n 2 2. - n - n Thus, r e s t r i c t e d jackknifing, being based on l i n e a r opera t ions , do-
es not induce any change over the o r ig ina l Ste in-rule es t imator in (2.4).
(11 ) Semi-restricted jackknifed estimator, tiere, jackknifing applied t o both the c l a s s i c a l es t imator Tn and t h e t e s t s t a t i s t i c
2 Pfi ( b u t not t o the o ther f a c t o r A,) , so t h a t D
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION?
"SS - "S ^SS where - - mn - (n-1 N n - l ( i ) , i = l , , , , , n , and - n , l
where T:-, ( i ) i s based on (X1 ,. . . ,X ,;i+l ,* * . ,:n) , f o r i = 1 ,... , -i-1 "S n . This i s a nonlinear adapta t ion, and, i n genera l , O n and e:SJ a re
not the same. (111) Unres t r i c t ed jackkni fed e s t ima tor . In t h i s case , the
jackknifing i s incorporated i n t he usual way o f recomputing the
(shrinkage) es t imators from each of the n sub-samples of s i z e n -1,
so t h a t
S U "S "S U where !n,i = n$, - Cn-1 ) in - l ( i ) , i= l , . . . ,n, and
;su - 2 - - ) - l i n l i ( i 3 1 (2.11)
with the A,,,-l ( i , defined on the subset (El ,. . . ,Xi-l , X i + l ,.. . ,tn 1, . . f o r i = l , , . . , n . Again we have a n o n l i ~ e a r adapta t ion, and hence, in
'v genera l , o3 and gSUJ a r e not the same.
-I1 -I I
Side by s i d e , we introduce the jackkni fed d i s p e r s i o n ma t r i x
"SRJ - -1 n "SR "SRJ "SR ;SRJ) I - n - n !n,i - B n ) ( !n,i - In a
^SSJ "SS - SSSJ)' iSS - e ) ( on = i -n - n 9
lCn "SU - gSUJ SU GSUJ)' i= l ( !n , i -n I D n - .. n
9ur main i n t e r e s t l i e s in the study of the a s p p t o t i c p rope r t i e s of tliese jacknifed Ste in-rule es t imators and t h e i r est imated disper-
sion matrices. The ro le of these l a t t e r es t imators i s examined in
the context of the est imation of the ADF! of the proposed jackknifed versions of the Ste in-rule es t imator i n (2.41.
3. FIXED ALTERNATIYE ASYMPTOTICS ,FOR JACKKNIFLNG
i t f o l l o ~ s from Sen (1984) t h a t f o r any (fixed) e # 0 , t he es- OS ... -
t i~r ia tors En and !n a r e asymptotically r i sk-equivalent in the sense
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2 2 5 2 SEN
2 tlie convergence of the f i r s t order negative moment of Tn ensures
t h a t the r a t i o of the two r i s k s asymptotically converges t o l , i . e . ,
E~T; ' B + 1 + 0 as n j m
^S => limn., {pn[e -., , e , ) / pnC? , !)I = 1 . (3 .1)
,iote t h a t t h i s negative moment condit ion i s needed not f o r the r i s k -
cor,il~:~tatiori of the c l a s s i ca l es t imator X , but f o r t9e s !~r inkage -. n one . For o ther modes of asymptotic equivalence of these two e s t i -
iliators [ such as almost sure [a.s.) equivalence o r s tocl !as t ic one ) ,
t h i s condition nay be dispensed with. In f a c t , when tlie underlying
d . f . F has f i n i t e 4th order moments, using the f a c t t h a t f o r 8 # 0, - , -2 n converges a.s. t o 0 (as n -+ a) , we obta in from (2.3) and some
elementary inequal i ty t h a t
6y v i r tue of (2 .7 ) , we conclude t h a t (3 .1) and (3.2) a l so hold f o r
ii:RJI . llote t h a t f o r the computation of the RDR r e s u l t s ( as will
be made in the next s e c t i o n ) , we do not need t h e negative moment-
conuit ion in (2 .1) .
ilext, we note t h a t Fn = n - ' x n - X i=l-.n-1 ( i ) . Hence, by (2 .3) , (2 .8)
and (2.9), we have
Let c,, be the sigma-field generated by the unordered col l ec t ion
,... ,/; ) and ;; , j > 1 , f o r n > 1 , so t h a t en i s rlonotone -11 ,n+j - - -
nonincreasing. Also, l e t pn = p( " S . k > n) be the sigma-field /'kY , k Y - gerierateti by K S - k > n , f o r n > 2, so t h a t F, ccn and F, i s ,kY,I<' - - a l so nionotone nonincreasing. Then
Thus, we have
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WHITHER J A C K K N I F I N G I N STEIN-RULE ESTIMATION?
2 -1 - n 2 - 2 lx (n-l)T, {An(n ZiZl{l - TnTn-l(i) S n - l ( i ) ) 1
= (n-
+ A,E - Now, f o r every 8 # 0, - *
( n- 1
while, independently of 8, -
where A = ch (41). Further, note t h a t P - -
2 T n = T ( y n , S n ) , n ? 2 (3 .8 )
where Xn and Sn - a r e both U - s t a t i s t i c s and T( . ) i s a smooth funct ion.
Hence, proceeding as in Sen (1977) [See h i s (3 .20) ] , i t follows
f o r every E > 0 (and we l e t O<E<!). Simi la r ly ,
2 -2 - - ! c ) = O ( n - 1 + ~ ) E{(1 - T , , T n 4 ) ( 5 , 4 - E n ) , ,, a . s . , a s n + -.
(3.10)
Thus, the r i g h t hand s i d e ( r h s ) of (3 .5) i s 0(n-"') a . s . , a s
n - -, and t h i s ensures t h a t f o r every ( f ixed) O b f 0, - nil 6isJ - $ % I I + 0 a . s . , a s n -+ (3.11)
Final ly , note t h a t d n = ch ( Q - I S - ' ) i s a ismoath' (nonnegative ) p - -n
function of ( t h e s tochas t i c matrix ) Sn , so t h a t by appeal t o Sen
(1977), we have
n l a x l < i < n I 1 % - I l i ) - A n I = 0(n-'2 ) a , s . , a s n -. , (3.12) --
a n d , t he re fo re , proceeding as in (3.4) through (3.10) and using
(2.4), (2.13) and (2 ,11) , we obtain t h a t f o r every (fixed ) R f 0 , - - 1 ^SUJ "S n Z I / - O n / / -. 0 a . s . , as n -. . (3.13)
*
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2 2 5 4 SEN
I t may be noted t h a t f o r the asymptotic a.s. equivalence res-
u l t s in (3.2), (3.11) and (3.13), t he a . s , convergence r e s u l t i n
(3.6) plays a basic role. I f f3 = 0 ( o r i s c lose t o 0 in the usual - w
Pitman-sense ), then the r i g h t hand s ide of (3.6) blows o f f , and , as a r e s u l t , (3.11) and (3.13) may not hold. This c a l l s f o r the
need to study these asymptotic r e s u l t s from a d i f f e r e n t perspect i -
ve when the pivot and the t r u e parameter point a r e c lose t o each
o the r , and we shal l take up t h i s i ssue in the next sec t ion,
From (3 .2) , (3-11) and (3.13), we note t h a t i f the t rue para-
rtieler (point ) 0 d i f f e r s from the p ivot (here , 0) by any fixed -. amount , jackknifing may not lead t o any s i g n i f i c a n t change t o the
" s usual Ste in-rule es t imator !n . Further , i n the same setup , i: - and 1: a r e asymptotically r i sk-equivalent , so t h a t shrinkage does -n not lead toany asylvptotic improvement over the c l a s s i ca l e s t i ~ a t o r
SRJ too, Noting t h a t a l l the th ree d ispers ion matrix es t imators V_., , vSSJ and !.':UJ involve (jackknifed) U - s t a t i s t i c s , we may proceed as , n in Sen (1977) and claim t h a t f o r any ( f ixed) Fi # 0 , as n -) m , - -
SUJ vSRJ i , vSSJ L and Y n -n -n 5 9 (3.14)
so t h a t T r ( Q l n ) converges t o Tr(Qz), in p robab i l i ty ( a s n -t ) in w-
each of the three cases of jackknifing considered e a r l i e r , Fur ther ,
note t h a t even under the f i n i t e n e s s of the second order moments of
F, S -f X a , s . (as n -t ) , and hence, Tr(4(ln) + T r ( 0 ~ ) a , s . , a s -11 - w..,
n + a . tlence, t he re i s no gain in the asymptotic r i sk-eff ic iency
~ u e t o the usual shrinkage estimation o r jackknifing on the top of
t h a t , and the r i s k est imation in the jackknifed s i t u a t i o n s may de- mand more s t r inqen t moment condit ions than in the usual case. Thus,
thcrc sceriis t o be very l i t t l e ground in aCvocating jackknifing in si~rinkage e s t i ~ ~ l a t i o n in the large sample case when the t r u e param-
e t e r 0 i s d i f f e r e n t from the pivot. This f ixed-a l t e rna t ive asymp- u
t o t l c p ic ture a l so provides f i n i t e sample j u s t i f i c a t i o n s when the pardmeter point O i s away from the pivot . !+owever, f o r e c lose t o
.., *
tre pivot , we may derive more meaningful r e s u l t s through consider-
& t i o m of Pitman-type a l t e r n a t i v e s , a s will be done i n Section 4.
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION?
4. ASYNPTOTICS FOR PITEAN-TYPE ALTERNATIVES IKnl
For Pitman-type (local ) a1 t e r n a t i v e s , t he dominance of t h e Ste in-rule es t imator over the c l a s s i c a l one follows d i r e c t l y from
the general r e s u l t s on shrinkage U-s t a t i s t i c s in Sen (1984). I t i s
therefore of i n t e r e s t t o study the performance c h a r a c t e r i s t i c s of the jackknifed versions under the same asymptotic se tup, and, the-
reby, t o examine the ro le of jackknifing in Stein-rule est imation
theory ( t h i s cons t i tu t e s the main theme of t h i s s tudy) . We conceive of a t r i angu la r a r r ay ; n - > 11) of
row-wise i . i .d.r.v. ' s , where f o r each n( 2 1 ) , X n i has the d , f .
F(? - nm4h ) , f o r some ( f ixed) h E E ~ , p 2 3 , and F has the mean
vector O and a p.d, d ispers ion matrix C (both unknown). We denote .. .. t h i s sequence ( of Pitman-type a l t e r n a t i v e s ) by {Knl . I t may be noted t h a t the X n i - n-'h , i 2 1 , a r e i . i ,d.r .v. with mean 0 and - dispersion matrix ; , and hence, by (2.2), under Kn , the r i s k of X i s given by l'r(QZ), f o r every n 1. We denote,for l a t e r use, -n -.. t h i s constant r i s k by
-9 P p(h) = Tr(QZ) ..- ( = limn, pn(Tn;n 'A ) ) , - X E E . (4.1)
For the computation of the asymptotic r i s k of the Ste in-rule e s t i - "S mator sn in (2 .3) , we may note t h a t under IKn} , ii2 converges in
d i s t r i b u t i o n to X-2 where x2 stands f o r a pos i t ive r,v. hav- P , A P ? A
ing the noncentral chi-square d i s t r i b u t i o n with p degrees of f ree-
dom (DF) and noncentral i t y parameter A = ?'C-'h - - , while dn conver- ges a.s. t o 6 = ch (QC); however, these a r e not enough t o ensure
P -- the ~i~ornent-convergence r e s u l t needed to show t h a t (2.2) converges ( under {I(,} ) t o a l i m i t as n increases inde f in i t e ly . I f we assume t h a t under {Kn) ,
d n ~ i 2 converges in L1-norm t o -2 ' - X ~ , A ' (4.2)
then, by appeal to the general r e s u l t s i n Sen(1984) I s e e a l so Sen
and Saleh (1985, Sec. 3)], we obtain t h a t under lKnl , whenever F possesses f i n i t e moments of order r , f o r some r > 2,
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SEN
= Tr(Qr) ..,- - 2 c ( p - 2 ) 6 ~ ( ~ ; ? ~ ) + 2 2
c a A*E( $4,A ) + T ~ P - -1 - -1 )E(x~:~,~) I , (4.3)
where A* = X ' C -1 -1 -1
* - C A and c = l imn- cn ( i t i s assumed t h a t
t h i s l i m i t e x i s t s and c E (0, 2(p-2) ) ) . Even w i t h o u t t h e L1-conv-
ergence r e s u l t i n (4.2), one may c o n s i d e r t h e asympto t i c d i i t r i b u -
t i o n o f n"( i: - n'% ) (under i K n I ) and compute t h e r i s k f rom
t h i s asympto t i c d i s t r i b u t i o n . T h i s i s termed t h e ADR o f t h e S t e i n - "S r u l e e s t i m a t o r fin . Then, as i n Sen (1984) w i t h a d a p t a t i o n s f rom
Saleh and Sen (1985), we conclude t h a t (4.3) ho lds f o r t h e ADR o f
;S , w i t h o u t r e q u i r i n g (4.2); f i n i t e n e s s o f t h e moments o f o r d e r r -. n o f F, f o r some r > 2 s u f f i c e s f o r t h i s . Now, t h e r i g h t hand s i d e
o f (4.3) i s l e s s t h a n T r ( Q C ) = p(X) f o r every f i n i t e A E E~ and c: -.., O < c < 2(p-2), and as A moves away from t h e o r i g i n ( i.e,, A o r
A* i nc reases ) , pS(X) converges t o p (A) .
Given t h i s asympto t i c p i c t u r e , i t i s o f i n t e r e s t t o s tudy t h e
e f f e c t o f j a c k k n i f i n g 011 t h e ADR ( o r t h e asympto t i c r i s k i t s e l f )
o f t h e S t e i n - r u l e e s t i m a t o r . A lso, t h e r e i s a more i m p o r t a n t ques- S t i o n : Can p (A) ( o r t h e o t h e r r i s k ) be es t imated c o n s i s t e n t l y by -
any o f t h e t h r e e j a c k k n i f i n g procedures ?
F i r s t , c o n s i d e r $,RJ. By ( 2 . 7 ) , we conclude t h a t (4.3) h o l d s
f o r t h i s r e s t r i c t e d j a c k k n i f e d v e r s i o n as w e l l . F u r t h e r , by (2 .5 ) ,
( 2 .6 ) and (2.12) , we have
Note t h a t under {Kn I ,
There fo re , by (4.4) and (4 .5 )
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION? 2257
and, as a r e s u l t ,
and t h e r h s o f (4 .7 ) i s a nondegenerate r . v . f o r every A: 11 < w.
Comparing (4 .3 ) and (4 .7 ) , we conclude t h a t whereas t h e r e s t r i c t e d - S j a c k k n i f e d e s t i m a t o r !:RJ agrees w i t h ", t h e a l l i e d j a c k k n i f e d
d i s p e r s i o n m a t r i x , v : ~ ~ , f a i l s t o p r o v i d e a c o n s i s t e n t e s t i m a t o r S o f t h e asympto t i c r i s k ( o r ADR) p ( A ) = pSRJ(h) . I n pass ing , we
may remark t h a t f o r Q = z-I, (6 = 1); we have ;he r h s o f (4 .7 )
equal t o p ( l - - 2 T2, and t h i s may even exceed p whenever - 2 C X ~ ,A
ck . 2 (wh ich has a p o s i t i v e p r o b a b i l i t y ) . Reca l l t h a t i n t h i s P,*
case o f r e s t r i c t e d j a c k k n i f i n g , we have a l i n e a r a d a p t a t i o n , so
t h a t t h e p i c t u r e w i t h t h e b i a s of t h e S t e i n - r u l e e s t i m a t o r remains
t h e same, b u t t h e o t h e r dual purpose o f e s t i m a t i n g t h e asympto t i c
r i s k ( o r t h e ADK) i s n o t served . Hence, t h e r e seems t o be n o t
n~uch o f impor tance i n t h e use o f t h i s r e s t r i c t e d j a c k k n i f i n g i n t h e
S t e i n - r u l e e s t i m a t i o n theory .
Consider n e x t t h e case o f a:SJ. We r e w r i t e ( 3 .3 ) as
t h a t
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2258 SEN
A t t h i s stage, we n o t e t h a t as n +
I Itn\ I = 0 (1 ) a.s.,
max l r i r n 1 l!i - !nl
(where we assume t h e f i n i t e n e s s o f 4 t h o r d e r moments o f F ) , and
under i K n ) (as w e l l as Ho: h - = O), - n f l I* , - 1 = 0 1 There fo re , -P
we o b t a i n ( t h r o u g h some s tandard s teps) t h a t
2 - 2 - - x 1 I =,,I = Z T ; ~ % + Ep(n-'t). E{ U i - 1 - Tn ) Un -n (4.15)
B Y ( 4 . 8 ) , (4.14) and ( 4 . 1 5 ) ~ we o b t a i n t h a t under {Kn l and t h e ass-
enled f i n i t e n e s s o f 4 th o r d e r moments o f F Censuring (4,1211, f o r
t n e s e n i i - r e s t r i c t e d j a c k k n i f i n g
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION?
"S TIIUS, i:SJ and !n a r e n o t a s y m p t o t i c a l l y e q u i v a l e n t under it$,},
though a s y m p t o t i c a l l y i:SJ a c t s l i k e a sh r inkage e s t i m a t o r ( w i t h
~ ~ 1 ; ~ b e i n g r e p l a c e d by ( p - 2 ) b n ~ i 4 ). T h i s suggests t h a t t h e usual
tecnn iques f o r e s t a b l i s h i n g t h e (asympto t i c ) dominance o f s h r i n k -
age e s t i m a t o r s o v e r t h e c l a s s i c a l ones may be used here t o d e r i v e
a p a r a l l e l r e s u l t f o r g;SJ. F o r t n e asympto t i c r i s k , we need t o
s t r e n g t h e n (4.2) t o
d n ~ i 4 converges (under Kn) i n L1-norm t o 6x-4 P,A '
(4.17)
Proceeding then as i n Sen (1984), we have under { K n l ,
where c = limn, cn , 6 = ch p (QL), -.- W - -. X ( w , I ) , u = , T - 1 2 ~ and A - - - .. -. = )>-'LQ-l y%
- - - - . F o r t h e p a r a l l e l ADR r e s u l t , we do n o t need (4,17);
f i n i t e n e s s o f t h e 4 t h o r d e r moments o f F ( e n s u r i n g (4 .12) ) and t h e
asympto t i c d i s t r i b u t i o n a l r e s u l t s on t h e component s t a t i s t i c s l e a d
us t o t h e same express ion ,
LfJe i n t e n d t o show t h a t f o r s u i t a b l e range o f c and o f p , t h e
rhs o f (4.18) i s s m a l l e r t h a n Tr(QC) ,ensur ing t h e asympto t i c domi- -- nance o f g:SJ o v e r Tn. Towards t h i s , we make use o f t h e S t e i n i d e -
n t i t y [ v iz . , Appendix B o f Judge and Bock (1978)J and o b t a i n t h a t
W ~ I W -.. -.-. 1 = AE( 1
= E( - ( P - ~ ) E ( ) , (4.19) - 6 -8
*E( xp:4,A = E( xp,* - PE( xpt2,* i (4.20) -1 -1 E~(W'W)-~W'AWI -. -. -. -.- = Tr(4 - - Z ) E C W , , ~ ~ , ~ ) + A*ECX~;~,*), (4-21 )
1 1 1 where A * = A ' C - 4- E- h , and i n o r d e r t h a t (4,21) i s w e l l de f ined , - - .-. - - "S we need t h a t p 2 7 ( compared t o p > 3 f o r !n ). I n t h i s c o n t e x t , -
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2260 SEN
we may note t h a t E ~ ( w ' u ) - ~ w ' A w ~ _.. _ _ _ - > ch ( A ) E ~ ( w ' w ) - 3 ~ , and hence, f o r P - .. -
the f in i t eness of the l e f t hand s ide of (4,21), p > 6 i s necessary
( s ince t h i s negative 4th moment does not a r i s e with the or ig inal
Ste in-rule es t imator , t h i s condit ion i s not necessary t h e r e . ) We
may fu r the r observe t h a t 6A*/6 - < O C I ~ ~ ( Q - ~ L - ~ ) - - = 1 and ~ T ~ ( Q - ' Z - ' ) - - -1 -1 -1 -1
= Tr(Q -. - T: ) c 1 ) 2 p. Thus, from (4.18) through (4 .21) , we
obtain t h a t
If we have t o have the asymptotic dominance of the jackknifed e s t -
imator gSSJ over Tn , we need t o have the inequal i ty sign in the -n l a s t formula holding f o r a l l A - z0, and , t h i s leads t o
p - > 7 and 0 < c < 2(p-4)(p-6)/(p-2)
= pSSJ(?) < p(?) , f o r every h E E'. - (4.23)
ye niay note f u r t h e r t h a t (p-4)(p-6) / (p-2) < (p-6) (p -2 ) , so t h a t
f o r the asyn~ptot ic dominance of t l ~ e semi-res t r ic ted jackknifed es-
t imator , we not only need p to be g rea te r than 6 , but a l s o c has
t o be r e s t r i c t e d t o a smaller domain ( which genera l ly leads t o a
s n ~ a l l e r reduction in the r i s k due t o shr inkage) . Also, in (4 .3 ) ,
i dea l ly , we take c = p-2 , while, in (4.18) o r (4 .22) , we would
have a smaller value f o r the optimal c. The question may na tu ra l ly
dt - ise ill t h i s context whether i:SJ dominates over $ in the l i g h t
oi' irle Airf? ( o r ttie asymptotic r i s k i t s e l f ) ? TO address t h i s que-
s l i u i ~ , we note t h a t by (4.3) ail0 (4 .18) ,
I t i s easy to show t h a t the f i r s t term on the rhs o f (4.24) i s non-
nevative , f o r a l l A ( i , e . , A > 0 ) . A t the null point ( i . e . , X = - S - d ) , the second termwis a l so nonnegative. Thus, pSSJ(0) p (c ) ,
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION? 2261
f o r a l l admissible values of c , and hence, t he asymptotic domina-
nce i n question does not hold, For h away from 0 , the second term -, ,"
on the rhs of (4.24) i s negative though i s small compared t o the
f i r s t term, so t h a t (4.24) i s genera l ly pos i t ive and q u i t e small , indica t ing t h a t the re i s no material change i n the A D R due to the
semi-res t r ic ted jackknifing over the Ste in-rule es t imator when the t rue parameter point i s d i f f e r e n t from the pivot (even i n t he local sense). Having t n i s naive p ic tu re on the ADR of the semi-res t r ic ted jackknifed Ste in-rule es t imator , we now proceed on t o examine tne
S otner important question : Can p (A) o r pSSJ(h) be estimated in a consis tent manner i n t h i s semi-restr icted jackknifing ? In t h i s se tup, X , being unknown, i s t r ea t ed as a nuisance parameter. ..
2 2 - Let us denote by "i = T i - T , G n i = - X n .
-1 n n i = l , ..., n , and l e t ! = n z i = l n n i . Note t h a t C i , l $ n i = O . Then, n using (4.11) through (4 .15) , i t follows by some standard s t eps
t h a t under {Knl and the assumed regu la r i ty condi t ions ,
Therefore, using (2 .13) , ( 2 . 8 ) , ( 2 . 9 ) , ( 3 . 3 ) , and (4 .13) - (4 .15) ,
along with (4.25) and (4 .26) , we obtain by some rout ine s t eps t h a t
under CKnl and the assumed regu la r i ty condi t ions ,
As a r e s u l t , we have
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Again, t h e r h s o f (4.28) i s a non-degenerate r . v . , and hence, h e r e
a lso , j a c k k n i f i n g f a i l s t o p r o v i d e a c o n s i s t e n t e s t i m a t o r o f t h e
asympto t i c r i s k p S S J ( ~ ) . +SUJ F i n a l l y , l e t us c o n s i d e r t h e case o f 8, . By (2.8) th rough
(2.11) , we have on d e f i n i n g nni, 6ni as i n be fo re ,
C = c n n o b t a i n
I
[ A t t h i s s tage, we assume t h a t IC, -~ - c n l = o(n- ') ; i n v iew o f
t h e f a c t t h a t limn-cn=c e x i s t s , we may always choose t h e sequence
tcn1 i n such a way t h a t t h i s ho lds. Suppose t h a t i n ( 2 . 4 ) , we
o n l y r e p l a c e cn by cnql and d e f i n e t h e r e s u l t i n g q u a n t i t y by R:. Then I \An - 1 = o ( n - i ) a.s., and t h i s rep lacement w i l l make no asympto t i c d i f fe rence . As such, f o r s i m p l i c i t y o f p r o o f , we l e t
- 1 = c.] Then, by an appeal t o [ (3 .13) o f ] Sen (1977) , we
t h a t as n -+ m ,
= I l n 1 - in I $ 1 1 I = ~ ( n - ~ ) ass., (4 .30)
where we make use o f t h e f a c t t h a t t h e elements of Sn a r e a l l
U - s t a t i s t i c s and d n = C h (QS ) i s a smooth f u n c t i o n o f 2,. F u r t h e r , P --n
n o t e t h a t (n-l)Ani = - (Xi - X,), l < i < n , so t h a t - -
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION? 2263
n i a ~ { \ ( n - , 1 ) 6 ~ ~ / : l < i < n - - 1 = o(ng ) a .s . , a s n - -. Further , by using (3.7) and C3.0) of Sen Cl977), we obtain t h a t
t , , Also, under {Knl , n 2 1 (x,,( / = 0 (1) . F ina l ly , using (4.13), we ob-
P t a i n t h a t
ma x -2 - 2 -% 1 T - T I 1 = oin ) a . s . , as n + . (4.12)
Consequently, from (4.29) through (4.32), we have under iKn) , ?- 6SSJ 1 + 0, in p robab i l i ty , a s n -+ . (4.33) n 2 1 1 $, --,,
As a r e s u l t , (4.16) through (4.23) a l so per ta in t o $UJ . Again , we may use (4.25) through (4.28) and conclude t h a t f o r t h e unrest- r i c t ed jackknifed version too, (4.27) and (4.28) hold. Thus, l i k e
the case of the semi-res t r ic ted jackknifing, here a l so , the dual
ro l e of minimaxity ( in the l i g h t of the ADR) and e s t i m a b i l i t y of
the ADR i s not s a t i s f a c t o r i l y served . 5. SOME GENERAL REMARKS
We have observed in Section 3 t h a t in the conventional f ixed d l t e rna t ive case, l i k e the Ste in-rule es t imator , i t s jackknifed
versions a r e of no special use i f n i s large. This asymptotic pic-
tu re a l so ind ica te s the performance c h a r a c t e r i s t i c s , f o r f i n i t e 11 , when 0 i s not very c lose t o the pivot (here, 0) . I f F i s i t s e l f
w .-. a rnultinormal d . f . , then, of course, such a f inite-sample study car1 be nade r e l a t i v e l y more p rec i se ly , but a t the cos t of tremend- ous mathematical complications. The asymptotic p ic ture w i t h the versions of jackknifing considered here a l so per ta ins t o the case of multi-normal F, and hence, suggests the lack of u t i l i t y of these
jackkriifeu versions f o r l a rge sample s i z e s when the p ivot and the t rue parameter points a r e not the same.
In the case of Pitman-type a l t e r n a t i v e s ( which i s q u i t e app- ropr i a t e i n t he asymptotic case ) , i t appears t h a t whereas $:RJ and '5 0 a r e in agreement, t he o ther two jackknifed versions a r e not so , , n
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;Yoreover, t h e o t h e r two v e r s i o n s have
f o r va lues o f p 2 7 C compared t o p > - c < 2(p-4) (p-6)/(p-2) << 2[p-2). From
asympto t i c dominance o v e r zn "S 3 f o r !n ) and f o r va lues o f
t h i s p e r s p e c t i v e , j a c k k n i f i n g
i s n o t ve ry a t t r a c t i v e . On t h e t o p o f t h a t , f o r e i t h e r of t h e
t h r e e j a c k k n i f e d ve rs ions , t h e co r respond ing j a c k k n i f e d e s t i m a t o r
o f t h e d i s p e r s i o n m a t r i c e s f a i l s t o p r o v i d e a c o n s i s t e n t e s t i m a t o r
o f t h e ADR. T h i s asympto t i c p i c t u r e a l s o p r o v i d e s a good i n d i c a t i -
on o f t h e f i n i t e sample s i z e s i t u a t i o n when i s ' c l o s e t o ' t h e
p i v o t . Thus, i n e i t h e r case, we have i n s u f f i c i e n t ground t o advoc-
a t e t h e use o f j a c k k n i f i n q i n S t e i n - r u l e e s t i m a t i o n .
For improved e s t i m a t i o n o f t h e niean v e c t o r o f a m u l t i n o r m a l
d i s t r i b u t i o n w i t h a known (p.d,) covar iance m a t r i x C , from c o n s i - - t i e r a t i o n s o f m i n i m a x i t y , Berger(1975) and Hudson(1974) b o t h sugge-
s t e d t h e use o f t h e f o l l o w i n g shr inkage e s t i m a t o r ( where a lso , we
use t h e p i v o t 0 ) : -
where
and r ( . ) = { r ( z ) , z E [O,m)} i s a s u i t a b l e nondecreasing and d i f f -
e r e n t i a b l e f u n c t i o n on [O,..); t h e y advocated t h e use o f t h e upper
bound o f 2(p-2) f o r r ( z ) . For p o s s i b l y unknown F and/or unknown Z,
one may r e p l a c e i n (5.1) and (5.2) Z by 2, , and cons ider a para-
l l e l e s t i m a t o r ; however, t h i s may n o t have t h e ( e x a c t ) m i n i m a x i t y
p r o p e r t y . The asympto t i c m i n i n l a x i t y o f such an e s t i m a t o r f o r Pitman
type l o c a l a l t e r n a t i v e s may be proved a long t h e same l i n e as i n
Sen (1984). As such, one may be i n t e r e s t e d i n i n c o r p o r a t i n g e i t h e r
o f the proposed j a c k k n i f e d v e r s i o n s i n t h i s S t e i n - r u l e e s t i m a t i o n .
By m a n i p u l a t i o n s v e r y s i m i l a r t o those i n S e c t i o n s 3 and 4, i t can
shown t h a t t h e genera l c r i t i c i s m s made on j a c k k n i f i n g i n e a r l i e r
s e c t i o n s a l s o p e r t a i n t o t h i s case. Thus, even i n such a somewhat
more genera l setup, we may n o t have s u f f i c i e n t grounds t o advocate
t h e use of j a c k k n i f i n g i n t h e S t e i n - r u l e e s t i m a t i o n theory. I f o u r S goal i s t o e s t i m a t e p (A), Sn may be used f o r L and we may a l s o
* .-" .-"
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WHITHER JACKKNIFING IN STEIN-RULE ESTIMATION?
use the following es t imators of A and A* :
1 1 while K and Tr(Q- Y- ) may be cons i s t en t ly est ivatec! by d n and - - ~ r ( ( j - ' ? ; ~ ) , respect ively . These may then be used in (4.3) t o pro- viut! the desired answer, Ilowever, i n t h i s context, we may note t h a t
under {1$,1 , the es t imators in (5.3) and (5.4) a re not cons i s t en t
( but have 1 iniiting non-degenerate d i s t r i b u t i o n s ) , and hence, t h i s
s i r ~ ~ p l e procedure may not serve the purpose wel l , Perhaps, jackkni-
fed est imators of O and A* wil l have b e t t e r performance in t h i s
context.
ACKNOWLEDGEMENTS
This work was p a r t i a l l y supported by the liational Heart , L u n g
and Blood I n s t i t u t e , Contract IJIH-IIHLRI-71-2243-L from the National
I n s t i t u t e s of tlealth. Thanks a re due t o the r e fe ree f o r h i s very
c r i t i c a l reading of the manuscript and useful comments.
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