An empirical Bayes approach to multiple decision procedures

Re-Bin Rau, Purdue University

Abstract

The problem of selecting the population with the largest mean from among $k({\ge2})$ independent populations is investigated. The population to be selected must be as good as or better than a control. It is assumed that past observations are available when the current selection is made. Accordingly, the empirical Bayes approach is employed. Combining useful information from the past data, empirical Bayes selection procedures are developed. In Chapter 2 and Chapter 3, we consider k independent normal populations with normal prior distributions. A single-stage empirical Bayes selection procedure is developed in Chapter 2. Then, this single-stage empirical Bayes selection procedure is extended to a two-stage empirical Bayes selection procedure in Chapter 3. The empirical Bayes selection procedures are proved to be asymptotically optimal, having a rate of convergence of order $O({({\rm ln}\ n)\sp2\over n}),$ where n is the number of past observations at hand. Chapter 4 deals with the problem of selection of the best from k independent Bernoulli populations. Here beta prior distributions are used. Two-stage empirical Bayes selection procedures are derived and studied. It is proved that the two-stage empirical Bayes selection procedures are asymptotically optimal, having a rate of convergence of order $O(\exp({-cn})),$ where c is a positive constant and n is the number of past observations at hand. Simulation studies are also carried out to investigate the performance of the proposed empirical Bayes selection procedures studied in Chapters 2, 3 and 4. These simulation studies were done for small to moderate values of n. The results indicate that the asymptotic optimal behavior of the selection procedures holds.

Degree

Ph.D.

Advisors

Gupta, Purdue University.

Subject Area

Mathematics

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