Unimodality of the Distribution of an Order Statistic

Abstract

A distribution function $G(x)$, on the real line, is called unimodal if there exists a value $x = a$, such that $G(x)$ is convex for $x < a$ and concave for $x > a$. Given that $G(x)$ is unimodal, a condition is given for the unimodality of $G^r(x)$, where $r$ denotes a positive integer. $G^r(x)$ represents the distribution function of the largest observed value in a sample of $r$ observations from the distribution $G(x)$. Some of the standard distributions, such as, the normal, gamma, Poisson and binomial distributions satisfy the given condition. An application of the given result to a problem of estimating the largest parameter is given.