Chapter 3 - Discrete Lifetime Models

Abstract

This chapter is concerned with the current and prospective tools that enable the representation of lifetime data through some model. A major component of such a formulation is to identify the probability distribution of the underlying lifetime. We discuss some possible lifetime distributions and their properties in this chapter. Often, a family of distributions is initially chosen and then a member that adequately describes the features of the data is assumed to be the model. We begin with the Ord family comprising of the binomial, Poisson, negative binomial, hypergeometric, negative hypergeometric and beta-Pascal distributions. The power series family and its subfamily, the Lerch distributions, are considered next. Among the distributions discussed in this class are the Hurwitz zeta, Zipf, Zipf-Manelbrot, Good, geometric, uniform, discrete Pareto, Estoup, Lotka, logarithmic and the zeta models. This is followed by considering the Abel series family consisting of generalized Poisson, quasi-binomial I, quasi-negative binomial, quasi-binomial II, and quasi-logarithmic series distributions. Further, the Lagrangian family, with special reference to the Geeta and generalized geometric models, are also discussed. The reliability characteristics of each of the above families such as hazard rate, mean residual life, characterizations based on relationships between reliability functions, recurrence relations, etc. are discussed. Of special interest in reliability modelling in discrete time is the development of discretized versions of acclaimed continuous life distributions. In this category, we review the works on discrete Weibull I, half-logistic, geometric, inverse-Weibull, generalized exponential, gamma and Lindley models and their reliability aspects. We conclude the discussions by presenting other models that do not belong to the above classifications like the discrete Weibull II and III and the S-distributions.