On some statistical inference problems using empirical Bayes approach

The empirical Bayes approach was introduced by Robbins (1956, 1964). Since then, it has become a powerful tool in statistical decision making. In this thesis, we consider several statistical inference problems using the empirical Bayes approach. First a decision problem of selecting good or better treatments compared with a control from k (≥2) treatments in the positive exponential family is considered in Chapter 2. A nonparametric empirical Bayes selection procedure is constructed and it is shown that the regret risk of the procedure goes to zero with a rate of [Special characters omitted.] , where n is the number of accumulated past observations at hand. In Chapter 3, the hypothesis-testing problem in the exponential family is studied. The kernel sequence method is introduced and used in the construction of a monotone empirical Bayes test. The asymptotic property of the proposed test is analyzed. Also a lower bound of monotone empirical Bayes tests is obtained. The estimation problem in the exponential family is considered in Chapter 4. The best possible rate of empirical Bayes estimators is obtained by converting the global problem into a local problem, identifying a functional of the marginal density, and then constructing the hardest two-point subproblem. An empirical Bayes estimator with rate close to this rate is constructed. Chapter 5 deals with the testing problem for a normal mean. The optimal rate of monotone empirical Bayes tests is obtained. This answers the question raised recently by Karunamuni (1996) and Liang (2000). In Chapter 6, the empirical Bayes approach is applied to a nonexponential family. The testing problem for an upper truncation parameter is studied. The optimal rate of monotone empirical Bayes tests is obtained and a test with this optimal rate is constructed.