Abstract : Let P sub 1,..., P sub k be k approximately equal to 3 given normal populations with unknown means theta sub 1,..., theta sub k, and a common known variance sigma squared. Let X sub 1,..., X sub k be the sample means of k independent samples o sizes n sub 1,...,n sub k from these populations. To find the population with the largest mean, one usually applies the natural rule d sub N, which selects in terms of the largest sample mean. In this paper, the performance of this rule is studied under 0 - 1 loss. It is shown that d sub n is minimax if and only if n sub 1 = ...= n sub k. d sub N is seen to perform weakly whenever the parameters theta sub 1,..., theta sub k are close together. Several alternative selection rules are derived in a Bayesian approach which seem to be reasonable competitors to d sub N, worth comparing with d sub N in a future simulation study.
For a sequence of statistical experiments with a finite parameter set the asymptotic behavior of the maximum risk is studied for the problem of classification into disjoint subsets. The exponential rates of the optimal decision rule is determined and expressed in terms of the normalized limit of moment generating functions of likelihood ratios. Necessary and sufficient conditions for the existence of adaptive classification rules in the sense of Rukhin [Ru1] are given. The results are applied to the problem of the selection of the best population. Exponential families are studied as a special case, and an example for the normal case is included.
Abstract : A natural class of two-stage procedures is proposed which can be completely described and studied in terms of Neyman-Pearson testing theory, where the unsymmetry of tests, however, can be overcome to a considerable extent. As a typical result it is shown that optimality of tests carries over to optimality of two-stage procedures. Finally, under normality, comparisons are made in case of k = 2 with certain Bayesian procedures.
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