The Discrete Analogue of the Weibull G Family: Properties, Different Applications, Bayesian and Non-Bayesian Estimation Methods

Abstract

In this work, we propose and study a new discrete analogue of the Weibull class. Many useful properties such as ordinary moments; moment generating function; cumulant generating function; probability generating function; central moments and dispersion index are derived. Two special discrete versions are discussed theoretically, graphically, and numerically. The hazard rate function of the new family can be "upside down", "increasing", "decreasing", "constant", "J-hazard rate function" and "double upside down" and "increasing-constant". Non-Bayesian estimation methods such as the maximum likelihood estimation; Cramer-von Mises estimation; the ordinary least square estimation and the weighted least square estimation are considered. The Bayesian estimation procedure under the squared error loss function is also presented. The Markov chain Monte Carlo simulations for comparing non-Bayesian and Bayesian estimation are performed using the Gibbs sampler and Metropolis Hastings algorithm. The flexibility of the new family is illustrated by four real datasets. The new family (through two special discrete versions) provided a better fit than sixteen competitive distributions.