CLOSED SUBSET-SELECTION PROCEDURES FOR SELECTING GOOD POPULATIONS

The problem of selecting a subset containing all good treatments is considered. It is shown that under mild conditions the class of procedures deciding correctly with at least a prespecified probability P* is equivalent in some sense to the class of all consonant and coherent multiple tests at multiple level 1-P* for a certain system of hypotheses. The equivalence relationship can be used to construct the so-called closed subset selection procedures by applying the well-known principle of closed test procedures. These procedures are more powerful than their single-step counterparts, the so-called natural selection procedures, in the sense that the selected subset of a closed selection procedure is never larger but often smaller than the selected subset of the corresponding natural decision rule. For a sampling statistic having a Lebesgue density with location parameter, analytical results are given concerning the critical values necessary to carry out the closed subset selection procedure. The normal case with both known and unknown common variance is treated in detail.