Properties of Probability Distributions with Monotone Hazard Rate

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)/ 1 - F(x). 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.