Contributions to nonparametric selection and ranking procedures

Sayaji Namdeo Hande, Purdue University

Abstract

Decision-theoretic and classical formulations of the ranking problems in a nonparametric setup are considered. Classical procedures are proposed and studied. The following problems are investigated: The problem of ranking k populations in a nonparametric setup when the population $\Pi\sb1,\...,\Pi\sb{k}$ are characterized by functionals of the associated distribution functions $\theta(F\sb1),\...,\theta(F\sb{k}),$ where $\theta(F\sb{i})=\int\ g\sb{i}dF\sb{i},$ and $g\sb{i}$'s are known bounded functions, is considered. Minimax rules for general loss functions, and Bayes rules for some specific loss functions are obtained and approximate non-randomized minimax rules are proposed. Classical procedures for selecting the best under the indifference-zone approach and subset selection approach are obtained and approximate non-randomized rules are derived. Also a lower bound for the probability of a correct selection is given. The problem of partitioning k ($\ge$2) multinomial cells according to the values of the cell probabilities is considered. Bayes rules are obtained for a general class of loss functions. A sequence of parametric empirical Bayes selection rules is proposed and shown to be asymptotically optimal of order $O(e\sp{-cn}).$ Also the problem of selecting the most (least) probable cell and simultaneously estimating the associated probability of the selected cell is considered. For this latter problem, a sequence of parametric empirical Bayes rules is proposed and shown to be asymptotically optimal of order $O(n\sp{-1}).$ The problem of ranking treatments, when only pairwise comparisons of the treatments are available is investigated. The problem of selecting the best treatment, more generally the problem of partitioning k ($\ge$2) treatments is studied. Bayes rules are obtained for a general class of loss functions. A sequence of parametric empirical Bayes selection rules is proposed and shown to be asymptotically optimal of order $O(e\sp{-cn})$. The problem of selecting a population close to a control or a standard is investigated in a nonparametric setup. The measure of distance between two distribution functions is assumed to be the Kolmogrov-Smirnov distance function. The problems of selecting the distribution closest to the control under the indifference zone approach and the subset selection approach are investigated. Asymptotic results on the probability of a correct selection are obtained.

Degree

Ph.D.

Subject Area

Statistics

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