Design of Experiments for Selection from Ordered Families of Distributions

Abstract

Given $G \in \mathscr{S}$ a space of cumulative distribution functions and a weak (reflexive and transitive) order relation $\prec$ on $\mathscr{S}$, the subclass of $\mathscr{S}$ given by $\{F \in \mathscr{S} \mid F < G\}$ is called an ordered family of distributions. Suppose $\pi_1,\cdots, \pi_k$ represent $k$ populations with distributions only known to belong to some specified ordered family. The general problem is to design an experiment to select $\pi_i$'s having large (small) $\alpha$-quantities. A preferred population is defined to be any population with $\alpha$-quantile "near" the $t$th largest $\alpha$-quantile and a correct selection occurs if the subset of populations selected contains at least a prespecified number, $r$, of preferred populations. The design problem is solved for both fixed and random subset size selection procedures under star and tail ordering. Tables of the sample sizes required to guarantee prespecified minimum probabilities of correct selection are given for the case of selecting from continuous IFRA populations. Comparisons are made with the optimal procedure for selecting from exponential populations. Properties of the proposed rules are discussed.