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Optimal robustness for estimators and tests online

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1^'f l^flEW YORK UNIVERSITY
' COURANT INITiTUTi = LIBRARY
25tM«r«rSt. New York, N.Y. 10012




NEW YORK UNIVERSITY
COURANT INSTITUTE OF
MATHEMATICAL SCIENCES



IMM-NYU 334
FEBRUARY 1965



OPTIMAL ROBUSTNESS FOR
ESTIMATORS AND TESTS

ALLAN BIRNBAUM AND EUGENE LASKA



PREPARED UNDER
CONTRACT NONR 285(38)
NR 042-206/11/18/64



^ U



IMM-NYU-53^
February 19 65



New York University
Courant Institute of Mathematical Sciences



OPTIMAL ROBUSTNESS FOR ESTIMATORS AND TESTS

Allan Birnbaum
Eugene Laska



This report represents results obtained at the Courant
Institute of Mathematical Sciences under the sponsorship
of the Office of Naval Research under Contract No. Nonr 285(58).
Reproduction in whole or in part is permitted for any purpose of
the United States Government.

*

Research Facility, Rockland State Hospital, Orangeburg, N. Y.



NEW YORK UNIVERSITY
COURANT INSTITUTE - LIBRARY
is I Mercer St. New York, N.Y. 10012



C. 1



1. Introduction and summary . Tukey (1960,1962) has provided
a broad perspective for research in efficiency-robustness of
estimators, as well as an Important part of the knowledge avail-
able in this area. The present paper is intended to complement
these by supplying formulations of concepts, techniques, and ini-
tial results for optimally efficiency-robust estimators and tests
in several tsrpes of problems. Relations to Tukey' s investigation
are discussed in Section 2, with brief reference to the related
work of Huber (1964). Relations to the approach to robust esti-
mation of Hodges and Lehmann (1963) are discussed in Section 5«
The present approach may be described as a formal indexing
of alternative specifications (e. g. "shapes" of error-distribu-
tions ) by a nuisance parameter, and adaptation of admissibility
and related concepts and Bayes techniques of the Neyman- Pears on
and Wald theories to the estimation and testing problems thus
formulated. Specific problems for which new optimal efficiency-
robust estimators are given are: linear estimation of location
parameters (Section 2); rank tests and related estimators for
two-sample problems (Section 3); and unbiased estimation
(Section 4). A by-product included in Section 4 is a generaliza-
tion of Stein's (1950) characterization of locally-best unbiased
estimators to the class of adinissible unbiased estimators togeth-
er with the corresponding complete class theorem.



2. Linear unbiased estimation of location parameters . Let X

be a random variable with p.d.f. f ( (x-|x )/j-, A ), where the finite

2
variance 0, A ■— A • Foi* each

A we assiome that the density function is symmetric. Let

(X,,...,X ) denote n Independent observations on X, and let

Y = (Y,,...,Y ) denote the same observations ordered nondecreas-

Ingly. We consider the problem of estimation of m,, restricting

consideration to linear unbiased estimators (LUEs), that is, es-

n

tlmators of the form |j,* = > a.y. for which E(m,*{Y) ||j.,'r, A ) = \i

identically in |x, cr, and AG A* Estimators will be appraised
in terms of their variance functions var(M,*||x,.7pv). When /\ con-
sists of a single point (i. e. the shape is known), the problem Is
reduced to one solved by Lloyd (1952), who derived best linear un-
biased estimators (BLUEs) (of o" as well as ii, without the as-
sumption of symmetry made here). With A unknown, the problem
leads to considerations which are conveniently illustrated first
in the artificially simple case that A contains just two ele-
ments, and C^



k^ if T(Zj^^,...,z^j^) - C^



where



N



Tm - T^t(z) - "^(2^^, ...,z^^) - 2_^ ^Ni^Ni



■N "N^



^1 i=l



o c



^% log g(w.,e)



e^'i



th
and vjhere W. is the i smallest of the N observations. The

notation Eq^k(') indicates expects tion is taken under the assump-
tion that the x' s and y's are distributed according to g(x, 0)


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Online LibraryAllan BirnbaumOptimal robustness for estimators and tests → online text (page 1 of 4)