Parameter Estimation for Type III Discrete Weibull Distribution: A Comparative Study

The type III discrete Weibull distribution can be used in reliability analysis for modeling failure data such as the number of shocks, cycles, or runs a component or a structure can overcome before failing. This paper describes three methods for estimating its parameters: two customary techniques and a technique particularly suitable for discrete distributions, which, in contrast to the two other techniques, provides analytical estimates, whose derivation is detailed here. The techniques’ peculiarities and practical limits are outlined. A Monte Carlo simulation study has been performed to assess the statistical performance of these methods for different parameter combinations and sample sizes and then give some indication for their mindful use. Two applications of real data are provided with the aim of showing how the type III discrete Weibull distribution can fit real data, even better than other popular discrete models, and how the inferential procedures work. A software implementation of the model is also provided. 1. Introduction Most reliability studies assume that time is continuous, and continuous probability distributions such as exponential, gamma, Weibull, normal, and lognormal are commonly used to model the lifetime of a component or a structure. These distributions and the methods for estimating their parameters are well known. In many practical situations, however, lifetime is not measured with calendar time: for example, when a machine works in cycles or on demands and the number of cycles or demands before failure is observed; or when the regular operation of a system is monitored once per period, and the number of time periods successfully completed is observed. Moreover, reliability data are often grouped into classes or truncated according to some censoring criterion. In all these cases, lifetime is modeled as a discrete random variable (r.v.). Indeed, not much work has been done in discrete reliability. Generally, most reliability concepts for continuous lifetimes have been adapted to the discrete case; in particular, discrete analogues of continuous distributions have been introduced [1]. In this context, geometric and negative binomial distributions are the corresponding discrete alternatives for the exponential and gamma distributions, respectively. Yet, discrete lifetime distributions can be defined without any reference to a continuous counterpart. Bracquemond and Gaudoin [2] provided an exhaustive survey on discrete lifetime distributions. The (two-parameter) continuous Weibull distribution is one of the most widely used