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Unimodality of the Distribution of an Order Statistic

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. Given that G(x) is unimodal, a condition is given for the unimodality of Gr(x), where r denotes a positive integer. Gr(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.