A Note on Strong Unimodality of Order Statistics

Abstract

A distribution function F(x) is said to be unimodal if there exists a value x = a such that F(x) is convex for x a. It is said to be strongly unimodal (Hajek and Sidak 1967 and Ibragimov 1956) if the convolution of F with any unimodal distribution function is unimodal. Obviously a strongly unimodal distribution function is unimodal. A nondegenerate distribution function F is strongly unimodal if and only if it has a density f that is logconcave within some open interval (a, b) such that X c a < b c X and fb f(x) dx = 1. Such a density is also called a PF2 density (Karlin 1968, Barlow and Proschan 1975). For a list of typical examples of strongly unimodal densities, see Example 1, Section 3. In this note, we study the strong unimodality of order statistics. Alam (1972) points out the relevance of the unimodality of order statistics for the construction of shortest confidence intervals for the largest of several location parameters. Under some regularity conditions, distribution of an order statistic is known to converge to a unimodal distribution (Buehler 1965). It is thus not surprising that some order statistics are unimodal even though the population may be far from unimodal. See, for instance, the bimodal density of Example 3, where all the order statistics, beginning with samples of size 2, are unimodal. On the other hand, order statistics from unimodal populations need not be unimodal (Example 6). Alam (1972) shows that for a population with density function f the condition