Duality between multiple testing and selecting

Selection problems are considered for which it is possible to formulate a correct selection via a so-called correct selection indicator. The correct selection indicator can be used to define a related multiple hypotheses testing problem. It is shown that there exists a bijective mapping from the class of all selection rules which guarantee a specified confidence level P* for a correct selection into a subfamily of all multiple tests at multiple level 1 - P*. First of all, this correspondence may be of interest from a pure theoretical viewpoint. Moreover, methods developed for multiple hypotheses testing, such as the closure principle, sometimes allow considerable improvements of existing rules. Last but not least, the duality between multiple testing and selecting may yield a better understanding of the structure of the underlying decision problem. Three selection problems will be considered in more detail: subset selection of Gupta's type, best-or-all selection, and unrestricted subset selection. Application of the closure principle to Gupta's selection problem indicates that Gupta's rule cannot be improved. In fact, Gupta's rule turns out to be admissible in a certain sense. It is shown how the best-or-all selection problem can be treated by applying the duality results, and its connection to Bechhofer's indifference zone approach is pointed out in detail. The idea of stepwise multiple tests is adopted for the unrestricted subset selection problem. The primary task in the derivation of step-down procedures is the determination of least favorable parameter configurations with respect to certain hypotheses. The validity of the suggested step-up selection procedures depends on the validity of a conjecture concerning the critical values for these procedures. The step-up approach can be considered as a more informative extension of the best-or-all selection rule.