Approximate Selection of All Populations Better Than a Control by Bootstrap

The problem of selecting all populations whose means are larger (better) than that of a control population is investigated. We employ the subset selection approach of Gupta and Sobel (Annals Math. Stat., 29(1), 235-244 1958). In this paper, unlike most previous studies, our setup is nonparametric. In the process of developing methods, two novel concepts regarding selection rules are introduced. First, for a value $$ p\in (0, 1) $$ p ∈ ( 0 , 1 ) , the concept of p-approximateness is introduced as the probability requirement for selection rules in a nonparametric setup. Second, a populationwise selection rule is defined. We propose a selection rule of better-than-control populations when the functional forms of population distributions are unknown and population variances are unequal and unknown. The constant which appears in the proposed selection rule is estimated by bootstrap. Consistency of the bootstrap is proved under rather weak conditions and it is shown that the limit inferior of the probability of correct selection of the proposed selection rule is larger than or equal to a prespecified value $$ p\in (0, 1) $$ p ∈ ( 0 , 1 ) . As a result the proposed selection rule is shown to be p-approximate and approximately populationwise. Simulation studies show that the proposed method has good performance in terms of the risk measured by the mean number of incorrect inclusions and exclusions of populations.