SOME CONTRIBUTIONS TO EMPIRICAL BAYES, SEQUENTIAL AND LOCALLY OPTIMAL SUBSET SELECTION RULES (RANKING, BEST, COMPARISON)

TACHEN LIANG, Purdue University

Abstract

Selection and ranking problems in statistical inference arise mainly because the classical tests of homogeneity are often inadequate in certain situations where the experimenter is interested in comparing k ((GREATERTHEQ) 2) populations with the goal of selecting the best or selecting good populations. One of the main approaches for selection and ranking problems is through the subset selection formulation which was pioneered by Gupta (1956, 1965). Since then, especially within the last ten years, different frameworks have been developed under the subset selection formulation (see Gupta and Panchapakesan (1979, 1984)). In this thesis, some results on empirical Bayes rules, sequential subset selection procedures and locally optimal subset selection rules have been obtained. Chapter I deals with the problem of selecting good populations through the empirical Bayes approach. Two selection problems have been studied: selecting populations better than a control or a standard and selecting all good populations among k populations. For each problem, a nonrandomized Bayes rule is derived for a linear loss function. Based on this Bayes rule, a sequence of empirical Bayes rules for selecting the good populations is derived. In each problem the rate of convergence of the sequence of the empirical Bayes rules is also studied. Chapter II deals with the problem of selecting the best population through the sequential subset selection approach. We use a modified sequential probabilistic ratio test to construct sequential selection procedures to select a subset such that (a) a population is eliminated as soon as there is statistical evidence indicating that it is not best population, and (b) when the procedure terminates and a subset is selected, one can assert, at some prespecified confidence level, the following: simultaneously, the best population is selected and the measure of separation between each selected population and the unknown best population is bounded by some prespecified value. In Chapter III, we consider some selection problems based on ranks under joint type II censoring. Our goal is to derive locally optimal subset selection rules for selecting a subset containing the best population. Problems are formulated according to whether the sample sizes from the k different populations are equal or not. Some properties associated with the partial ranks under the joint type II censoring are given. Locally optimal subset selection rules R(,1) (for the equal sample sizes case) and R(,2) (for the unequal sample sizes case) are derived and some locally monotone properties of R(,1) and R(,2) are also discussed.

Degree

Ph.D.

Subject Area

Statistics

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