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On the conditional probabilities of correctly selecting the best. (English) Zbl 0802.62024

Summary: Suppose there are \(k\) populations different only in location parameters which are assumed to be unknown. To select the best population, the one having the largest location, the natural selection rule is to select the population giving the largest observation. The probability of correct selection achieved in such a selection has been extensively studied in the literature. However, if the selection of the best population is made only if some preliminary test is significant, then the probability of correct selection achieved in such a selection should be conditioned on the event that the test is significant.
We compare this conditional probability of correct selection with the classical unconditional probability of correct selection. We show that the conditional probability of correct selection can be different from the unconditional probability of correct selection in either direction, larger or smaller, even in some usual situations. We also show that if the preliminary test is the range test or a test based on the statistic \(X_{[k]}- X_{[k -1]}\) then the conditional probability of correct selection is always no less than the unconditional probability of correct selection under some distributional assumptions. Some unsolved problems are also discussed.

MSC:

62F07 Statistical ranking and selection procedures
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