## The Annals of Mathematical Statistics

### Properties of Probability Distributions with Monotone Hazard Rate

#### Abstract

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 $q$.

#### Article information

Source
Ann. Math. Statist. Volume 34, Number 2 (1963), 375-389.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177704147

Digital Object Identifier
doi:10.1214/aoms/1177704147

Mathematical Reviews number (MathSciNet)
MR171328

Zentralblatt MATH identifier
0249.60006

JSTOR