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.