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expected subset size risk; largest location parameter; Gupta-type rule; $L$-estimates; linear combinations of order statistics; monotone likelihood ratio; minimax; asymptotic normality
The problem of selecting a subset of polulations containing the population with the largest location parameter is considered. As a generalization of selection rules based on sample means and on sample medians, a rule based on $L$-estimates of location is proposed. This rule is strongly monotone and minimax, the risk being the expected subset size, provided the underlying density has monotone likelihood ratio. The problem of fulfilling the $P*$-condition is solved explicitly only asymptotically, under the asymptotic normality of the $L$-estimates used. However, after replacing their asymptotic variance by its estimate, the solution becomes distribution free.
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