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

Abstract

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.

Keywords:

• nonparametric subset selection
• unequal sample sizes
• quantiles

AMS 2000 Subject Classification::

• 62F07
• 62G99

Acknowledgements

Mr Conroy Lum, FPInnovations introduced the second author to the application that led to the work reported in this paper and we are indebted to him. The work was partially completed as part of a research programme funded by a Natural Science and Engineering Research Program Collaborative Research and Development Grant with contributions from FPInnovations. We thank two anonymous reviewers for suggestions that improved the expository quality of the paper.