On some empirical Bayes and statistical selection and ranking procedures

The dissertation deals with some empirical Bayes test procedures and statistical selection and ranking procedures. The first two chapters are related to empirical Bayes test procedures. In the first chapter, empirical Bayes tests in some general cases are investigated. Under the traditional empirical Bayes framework, the conditional distributions of observations at different stages are independent and identical. Only keeping the independence assumption, we develop a new general approach for constructing asymptotic optimal empirical Bayes tests for nonidentical cases as well as identical cases. Convergence rates of the constructed empirical Bayes tests are studied. In two special cases, the best convergence rates can be achieved for the empirical Bayes tests by this approach. Moreover, simulations for these two cases demonstrate prominent results. Chapter 2 applies the general approach in Chapter 1 to the double exponential distribution. A special relation between the prior distribution function and the marginal distribution function for the double exponential distribution is found and used to construct empirical Bayes tests for this distribution. The other two chapters are related to statistical selection. In Chapter 3, isotonic subset selection procedures for selecting populations better than a standard are investigated for the double exponential location parameter problem. These isotonic procedures are compared based on expected number of bad populations in the selected subset. All these selection procedures are optimal with respect to a special loss function. To develop selection procedures which are optimal for a class of loss functions, the Bayes P* subset selection procedures are proposed and verified to the problem of selecting populations better than a standard in Chapter 4. Implementations of these subset selection procedures to normal, binomial and Poisson distributions are presented