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NEW YORK UNIVERSITY

COURANT INSTITUTE OF

MATHEMATICAL SCIENCES

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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)