A discrete inverse Weibull distribution and estimation of its parameters

Abstract

A discrete analogue of the standard continuous Weibull distribution was proposed in the literature to meet the need of fitting discrete-time reliability and survival data sets. Its properties were studied and the methods of estimation of its parameters were also investigated by various authors. Analogous to its continuous counterpart, the discrete Weibull does not provide a good fit to data sets that exhibit non-monotonic hazard rate shapes. In this paper, we propose a discrete inverse Weibull distribution, which is a discrete version of the continuous inverse Weibull variable, defined as X − 1 where X denotes the continuous Weibull random variable. It is shown that the hazard rate function can attain a unimodal or monotone decreasing shape for certain values of parameters. We then proceed to study four methods of estimation (the heuristic algorithm, the inverse Weibull probability paper plot, the method of moments and the method of proportions). From the results of extensive simulation runs, their accuracies and precisions are compared. It is found that for right skewed discrete inverse Weibull distributions, the last two methods seem wanting due to certain characteristics of the estimation procedures and numerical convergence. The inverse Weibull probability paper plot and the heuristic method fare better. Finally, a discrete data set is fitted by both the discrete Weibull and the discrete inverse Weibull and their AICs are compared.