Subset selection – extended Rizvi–Sobel for unequal sample sizes and its implementation

A nonparametric procedure is presented for selecting a subset of a set of k populations, containing the one with the largest (L) or smallest (S) αth quantile when independent samples are available from each and one population is the uniformly correct choice whatever be α. The result, an extension of a method previously proposed for the case of equal sample sizes, includes population i, if its αth sample quantile exceeds (in the case of L) the largest of the sample (α−β)th quantiles for the other populations, where 0<β<α. The selection index β is specified by the user. An obvious adaptation of this rule covers S. An asymptotic theory for the method gives a practical way of selecting β by optimising a linear combination of the probability of correct selection, which ideally should be large, and the expected subset size, which ideally should be small. Furthermore, the criterion provides a way of selecting the sample sizes in situations where the cost of obtaining the samples differs for the different populations.