ON THE CHOICE OF COORDINATES IN SIMULTANEOUS ESTIMATION OF NORMAL MEANS

DIPAK KUMAR DEY, Purdue University

Abstract

Let X be a k-variate vector normally distributed with mean (theta) and known covariance matrix (SUMM). Given a set of coordinates, it is known what shrinkage type estimators to use. It is not known, however, what coordinates to use. In the case of estimating (theta), we consider the question of choice of coordinates, and call it the "separation problem", in that we often must decide whether to use all coordinates in one combined shrinkage estimator or separate into groups and use separate shrinkage estimators on each group. The "combined" and "separate" estimators generally have risk functions which cross, i.e., the performance of one estimator is better than the other in one region and worse in another region. Therefore, we will use criterion involving Bayes risks to decide whether or not to separate. We consider the robust generalized Bayes estimators, as in Berger (1980), in estimating the mean in both the "combined" and the "separate" problems. We first consider the case of accurately specified priors and show that, somewhat surprisingly, the combined estimator is better for normal priors and for various flat tailed priors. Then we show that for large k, asymptotic separation can be better for flat tailed priors. We next consider the question of inclusion of extreme observations, when a flat tailed prior is suspected. This question was first studied by Stein (1974), who obtained partial answers. We obtain for a broad class of flat tailed priors the optimum truncation point of a shrinkage estimator. Finally, we consider the situation in which part of the prior information may be "misspecified", corresponding to a situation in which certain coordinates have much less certain prior information than others. It is shown in such a situation that separation can be better, and a boundary in terms of the amount of misspecification is found.

Degree

Ph.D.

Subject Area

Statistics

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