Multiple decision theory: Ranking and selection problems

Multiple decision theory is concerned with those decision problems in which there are a finite number of possible decisions. The most widely used form of multiple decision theory argues that preferences among alternatives can be described by the maximization of the expected value of a numerical utility function, or equivalently, the minimization of the expected value of a loss function. Probability and statistics are usually heavily involved to represent the uncertainty of outcomes, and Bayes Law is frequently used to model the way in which new information is used to revise beliefs. An important branch within multiple decision theory is ranking and selection: how to select a statistical model or population according to some pre-determined criteria; and once a selection is made, how to investigate its performance. This dissertation tries to resolve some ranking and selection problems. First, a selection problem which originates from measurement error models is investigated in Chapter 2. A selection procedure is developed for selecting the treatment which has the largest regression slope, and the performance of the selection rule in terms of the probability of making a wrong decision is studied as well. In Chapter 3, a two-stage selection procedure for selecting the best Bernoulli population is studied using negative binomial sampling scheme under the Bayes and empirical Bayes framework. Then in Chapter 4, a problem for selecting the largest logistic population mean is investigated and the asymptotic optimality of the proposed selection rule is analyzed. In Chapter 5, isotonic subset selection procedures for gamma distribution family are investigated given prior information about the ordering. Chapter 6 is concerned with simultaneous selection and estimation procedures