On selecting a subset containing the population with the smallest variance

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Abstract
A multiple decision approach is taken to the problem of selecting a subset from k given normal populations which includes the ‘best’ population, i.e. the one with the smallest population variance. The population variances of the k normal populations are unknown and the population means may be known or unknown. Based on a common number of observations from each population, a procedure R is defined which selects a subset which is never empty, small in size and yet large enough to guarantee with pre-assigned probability that it includes the best population, regardless of what are the true unknown population variances. Expressions for the probability of a correct selection using R are derived and it is shown that, for the case in which the k sample variances have a common number, v , of degrees of freedom, the infimum of this probability is identical with the probability integral of the ratio of the minimum of k -1 independent chi-squares to another independent chi-square, all with v degrees of freedom. The associated distribution theory for this statistic and the tables needed to carry out the procedure R are given in a companion paper (Gupta & Sobel, 1962). Formulae are obtained for the expected number of populations retained in the selected subset, and it is shown that this function attains its maximum when the population variances are all equal. Two generalizations are considered; one deals with the case of unequal degrees of freedom and the other is concerned with a procedure for the selection of the t (1 ≤ t < k) best populations.

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