MULTIPLE DECISION PROCEDURES FOR TUKEY\u27S GENERALIZED LAMBDA DISTRIBUTIONS (SAMPLE MEDIANS, SELECTION, RANKING, ASYMPTOTIC, RELATIVE EFFICIENCIES, ISOTONIC, LOGISTIC)

Selection and ranking (more broadly multiple decision) problems arise in many practical situations since it is now well-recognized that the classical tests of homogeneity usually do not provide the answers the experimenter wants. In this thesis we study Tukey\u27s lambda distributions as the underlying model for selection and ranking problems. It is known that the family of Tukey\u27s generalized lambda distributions is very broad and contains most well-known distributions as special cases. Chapter 1 deals with selection and ranking problems based on sample medians for the symmetric lambda distributions and gives applications of the lambda family of distributions. We investigate some properties of the lambda family of distributions. We also propose some selection procedures and study the properties of these procedures. An application of the lambda distribution for approximating some constants used in the selection and ranking procedures for other symmetric distributions is made. In Chapter 2, the problems of isotonic selection procedures for the family of lambda distributions and for logistic distributions are considered. Some isotonic procedures are proposed and studied. The approximation of constants used in the proposed procedure is investigated. It is shown that the isotonic procedures are better than some classical procedures in terms of reducing the expected number of bad populations in the selected subsets. Chapter 3 deals with the problem of choosing the optimal score function for different nonparametric procedures proposed by Nagel (1970) and Gupta and McDonald (1970). A Monte Carlo study is carried out. It indicates that the score function based on uniform distribution is optimal and robust against possible deviations from the underlying distributions. In Chapter 4, a two-stage elimination-type procedure under the Bayesian setting is proposed and its properties are studied. In particular, we use a stopping rule to construct a 100(1-2(alpha))% Highest Posterior Density Credible region with a common width 2d for the unknown means of selected populations at stage 1. A Monte Carlo study is carried out to examine the performance of the proposed procedure