Asymptotic consistency for subset selection procedures satisfying the P-condition

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Abstract

This paper deals with the multiple decision problem of subset selection, restricting attention to procedures which control the probability that the best population is selected. For general classes of loss functions we consider asymptotic consistency, defined as the property that the risk of a procedure tends to the minimum loss as the sample size tends to infinity. Necessary and sufficient conditions are derived for both pointwise and uniform (on compact sets) consistency.

The general theory is applied to selection problems in normal, binomial, multinomial and multivariate normal populations. Some highlights are: For selecting normal means, Gupta's procedure is the only one in Seal's class that is consistent. Classes of minimax procedures for selecting multinomial cells are not consistent. For selecting multivariate populations according to the Mahalanobis distance the procedure based on maximum likelihood estimates is not consistent. Procedures proposed by Gupta and his coauthors are found to be, in most cases, consistent.

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