Some Bayes and Empirical Bayes Selection Procedures

摘要

A common problem faced by an experimenter is one of comparing several populations (processes, treatments). Suppose that there are K (< or = 2) populations pi 1,...., pi k and for each i, pi (i) is characterized by the value of a parameter of interest, say theta i. The classical approach to this problem is to test homogeneity hypothesis HO: theta 1 = ... = theta k. However, the classical tests of homogeneity are inadequate in the sense that they do not answer a frequently encountered experimenter's question, namely, how to identify the best population or how to select the more promising (worthwhile) subset of the populations for further experimentation. These problems are known as ranking selection problems. In the present paper, we describe selection and ranking procedures using prior distributions or using the information contained in the past data. Section 2 of this paper deals with the problem of selecting the best population through Bayesian approach. An essentially complete class is obtained for a class of reasonable loss functions. We also discuss Bayes-P* selection procedures which are better than the classical subset selection procedures in terms of the size of selected subset. In Section 3 we set up a general formulation of the empirical Bayes framework for selection and ranking problems. Two selection problems dealing with binomial and uniform populations are discussed in detail.