On Some Properties of Life Distributions with Increasing Elasticity and Log-concavity

The purpose of this paper is to extend and systematize known results in log-concave and log-convex properties of life distributions. Also, to discuss the closure property of increasing generalized failure rate (IGFR) distributions with respect to mixing operation. Mathematics Subject Classification: 62N05, 62G10 The study of the log-concavity and log-convexity are useful in many areas of economics, political science, biology, actuarial science and engineering. It is often important to make explicit assumptions on the underlying distribution. However, in some situations there is no closed form expression for the distri- bution functions, the failure rates, the mean residual lifetime (MRL), and the variance residual lifetime (VRL) and it is still of interest to study the proper- ties of such functions. Most of the existing results in the literature have dealt with the density functions, distribution functions, and their integrals. Some of these results have been related to reliability functions, failure rates, and MRL functions. Yet the log-concave and log-convex properties of the VRL have not been touched. For this reason, we study the log-concavity for the VRL. In Section 2, we give definitions and basic notations for continuous concave distributions. We also give some examples for common continuous distribu- tions. Section 3 discusses the log-concavity involving failure rates, MRL and VRL. In Section 4, we consider log-concavity for discrete life distributions. We discuss stochastic orderings that are related to concave distributions in Section 5. Finally, we study the property of IGFR distributions.