## The Annals of Mathematical Statistics

### 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 $\alpha = \frac{1}{2}$.

#### Article information

Source
Ann. Math. Statist., Volume 38, Number 6 (1967), 1788-1803.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177698612

Digital Object Identifier
doi:10.1214/aoms/1177698612

Mathematical Reviews number (MathSciNet)
MR221693

Zentralblatt MATH identifier
0214.46001

JSTOR
Rizvi, M. Haseeb; Sobel, Milton. Nonparametric Procedures for Selecting A Subset Containing the Population with the Largest $\alpha$-Quantile. Ann. Math. Statist. 38 (1967), no. 6, 1788--1803. doi:10.1214/aoms/1177698612. https://projecteuclid.org/euclid.aoms/1177698612