Open Access
December, 1972 Unimodality of the Distribution of an Order Statistic
Khursheed Alam
Ann. Math. Statist. 43(6): 2041-2044 (December, 1972). DOI: 10.1214/aoms/1177690881


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.


Download Citation

Khursheed Alam. "Unimodality of the Distribution of an Order Statistic." Ann. Math. Statist. 43 (6) 2041 - 2044, December, 1972.


Published: December, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0251.62031
MathSciNet: MR359175
Digital Object Identifier: 10.1214/aoms/1177690881

Rights: Copyright © 1972 Institute of Mathematical Statistics

Vol.43 • No. 6 • December, 1972
Back to Top