Nonparametric Procedures for Selecting A Subset Containing the Population with the Largest $\alpha$-Quantile

Abstract

A nonparametric solution is given for the problem of selecting a subset of $k$ populations which with high probability contains the one with the largest $\alpha$-quantile, $0 < \alpha < 1$. It is assumed that each population has associated with it a continuous cdf $F_i(x) (i = 1, 2, \cdots, k)$, that samples of equal sizes are taken from each population and that observations from the same or different populations are all independent. We use the notation of the companion paper [7] without redefining all the symbols and, in particular, it is convenient to identify each population by its cdf. A procedure for the problem, based on order statistics is proposed in Section 2 and limitations on its feasibility in terms of a bound $P_1$ on the possible guarantee probability $P^\ast$ are examined in Section 3. An asymptotic expression in terms of tabled functions is also given for $P_1$. The expected size of the selected subset is examined exactly in Section 4 and asymptotically in Section 5; the latter also contains an asymptotic formula for the smallest sample size $n$ required to meet the $P^\ast$-guarantee. Asymptotic relative efficiency evaluations of the proposed procedure are carried out in Section 5. Section 6 treats the dual problem of selecting the smallest $\alpha$-quantile; Section 7 verifies a monotonicity property. Extensive tables giving $P_1$ and the integer constant defining the procedure are provided for the case $\alpha = \frac{1}{2}$.