Aug 12, 2005 - The Reverse Stein Effect is identified and illustrated: A statistician ..... Jill j. (39). Because B]. 15 is a ball with 0 E 8(B] - the set. sptleJ'lcal cap on ...
Reversing the Stein Effect Michael D. Perlman and Sanjay Chaudhuri lJeva7't'ITI,ent of Statistics,
of
~vA
98195, USA
12 August 2005
Abstract
The Reverse Stein Effect is identified and illustrated: A statistician who shrinks his/her data toward a point chosen without reliable knowledge about the underlying value of the parameter to be estimated but based instead upon the observed data will not be protected by the minimax property of shrinkage estimators such a" that of James and Stein, but instead will likely incur a greater error than if shrinkage were not used. Key vVords and phrases: James-Stein estimator, shrinkage estimator, Bayes and empirical Bayes estimators, multivariate normal distribution.
Footnote: NSA .. ,
1
The Case for Shrinkage: the Stein Effect
Supp'Jse that X is an observed random vector in Euclidean such that X = Y 0, where 0 is an unknown location parameter and Y is an unobserved continuous random vector. Under the mild aSfmnJPtkm that Y X - 0 is directionally , it is easy to heuristi,c:ally "slrrirlkage" estimators for 0 of the form (0) .
where-y(X E 1) and 00 is any fixed shrinkage point in The improvement offen~d by such shrinkage estimators is often referred to as the Stein Effect. for fixed 0 and 00, let B 1 C lRlJ' denote the ball of radius 110 centered at 00 and let H be the halfspace bounded by a hyperplane aH tangent to B 1 at 0 (see Figure 1). Then
{X
I
- 0011> 110 - ooll} = Bf, 0011 > 110 00111 00] PI'S
PI'S [IIX
(2) [X E Bf
I
>Prs[XEHloo] 1 2'
(3)
where (3) follows from directional symmetry by Proposition l(c) in Appendix 1. Furthermore, under somewhat stronger but still general assumptions Proposition 2 in Appendix 1), lim PI'S [IIX
p-+oo
0011
0011 > 110 - 0011]
lim Prs [X E Bf !I 001"
p-+,oo
1.
is usually an overestimate of 110 - 0011, so an estimator of the - (0) for 0- 00 should be preferable to X - 00 itself. 00 immediately leads to estimators for 5 of the form Appendix 2),
{X
I
2
assuInlptionc of l'rclPOf,jtieln 2 in Appendix 1.
=1.
Figure 1: The balls B j and B 2 in (2) and (5).
2
The Stein Paradox
Assume now that Y '" N p (O,O'21 p ), the multivariate normal distribution with mean 0 and covariance matrix O'21 pl where 0'2 > 0 is known, so X", N p (5, O'21 p ). In this simple case, the James-Stein (.IS) estimator for 5 is given by (8)
where X (Xl,.. '" N p (5,O'21 p ) with 0'2 known and 50 is a fixed but arbitmry point in ~. The truncated == "plus-rule" .IS estimator
O'2(p ( 1-
Ii. v
ii./\.
2)) £ li2 (JO I
(X -
50)
(9)
is a estimator of the form (1). These renowned estimators have the property that when p ~ 3, they dominate X under both th(~ mean sqmlre error and Pitman closeness (PC) . for every fixed 6,50 E
(11)
and approaches 1 as p -+ 00, where denotes a noncentral chi-square random variate with p degrees of freedom and noncentrality parameter rj. Note especially that: the improvements offered by the .JS estimators can be great, especially when p is large: if 0 = 00 then MSE(6js) Vel.,,;:,\'}11.
. An appnJa(:h to the recovery of intm-bJock information. In J. F. N. Ed. New York.
G. S.
York.
