Open Access
June, 1963 Properties of Probability Distributions with Monotone Hazard Rate
Richard E. Barlow, Albert W. Marshall, Frank Proschan
Ann. Math. Statist. 34(2): 375-389 (June, 1963). DOI: 10.1214/aoms/1177704147


In this paper, we relate properties of a distribution function $F$ (or its density $f$) to properties of the corresponding hazard rate $q$ defined for $F(x) < 1$ by $q(x) = f(x)/\lbrack 1 - F(x)\rbrack$. It is shown, e.g., that the class of distributions for which $q$ is increasing is closed under convolution, and the class of distributions for which $q$ is decreasing is closed under convex combinations. Using the fact that $q$ is increasing if and only if $1 - F$ is a Polya frequency function of order two, inequalities for the moments of $F$ are obtained, and some consequences of monotone $q$ for renewal processes are given. Finally, the finiteness of moments and moment generating function is related to limiting properties of $q$.


Download Citation

Richard E. Barlow. Albert W. Marshall. Frank Proschan. "Properties of Probability Distributions with Monotone Hazard Rate." Ann. Math. Statist. 34 (2) 375 - 389, June, 1963.


Published: June, 1963
First available in Project Euclid: 27 April 2007

zbMATH: 0249.60006
MathSciNet: MR171328
Digital Object Identifier: 10.1214/aoms/1177704147

Rights: Copyright © 1963 Institute of Mathematical Statistics

Vol.34 • No. 2 • June, 1963
Back to Top