Properties of Probability Distributions with Monotone Hazard Rate

Abstract

In this paper, we relate properties of a distribution function F (orits density f) to properties of the corresponding hazard rate q definedfor $F(x) < 1$ by q(x) = f(x)/[ 1 - F(x)]. It is shown, e.g., thatthe class of distributions for which q is increasing is closed underconvolution, and the class of distributions for which q is decreasingis closed under convex combinations. Using the fact that q is increasingif and only if 1 - F is a Polya frequency function of order two,inequalities for the moments of F are obtained, and some consequencesof monotone q for renewal processes are given. Finally, the finitenessof moments and moment generating function is related to limitingproperties of q.