## The Annals of Mathematical Statistics

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

### Properties of Probability Distributions with Monotone Hazard Rate

Richard E. Barlow, Albert W. Marshall, and Frank Proschan

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### Citation

Barlow, Richard E.; Marshall, Albert W.; Proschan, Frank. Properties of Probability Distributions with Monotone Hazard Rate. Ann. Math. Statist. 34 (1963), no. 2, 375--389. doi:10.1214/aoms/1177704147. https://projecteuclid.org/euclid.aoms/1177704147