Chapter 5 Bathtub Distributions

In the last chapter, we considered models in which the hazard rate function and mean residual function were monotone. There are several practical situations wherein these reliability functions exhibit non-monotone behaviour. Important among them are distributions which have hazard rates that are bathtub and upside-down bathtub shaped periodic, polynomial type, roller-coaster shaped, etc. This chapter is devoted to the study of distributions possessing such hazard rates and their properties. We first consider bathtub-shaped hazard rates and non-monotone mean residual life functions and their inter-relationships. Various distributions studied in literature in this connection are reviewed. Data sets that manifest different types of behaviour warrants models that can accommodate and explain the special characteristics in them. With this general requirement in mind, we present various methods of constructing discrete bathtub models. Some theorems in this connection are proved first. This is followed by prescribing some methods that lead to bathtub and upside-down bathtub distributions. These include discretizing continuous bathtub models, use of mixtures, and convex functions. We also examine whether bathtub models possess closure properties with respect to various reliability operations such as formation of mixtures, convolution, coherent systems, equilibrium and residual life distributions. Further, we present definitions and properties of periodic hazard rates. Several examples of distributions are provided to illustrate the concepts, methods and properties discussed here.