The Weibull-Geometric distribution

TL;DR

This paper introduces the Weibull-Geometric distribution, a flexible model capable of capturing various failure rate behaviors, including unimodal shapes, with methods for parameter estimation and real data applications.

Contribution

The paper presents the Weibull-Geometric distribution, extending the exponential-geometric distribution, with derivations of its properties, estimation procedures, and practical applications.

Findings

Distribution can model unimodal failure rates

Maximum likelihood estimation algorithm provided

Applied to real data demonstrating flexibility

Abstract

In this paper we introduce, for the first time, the Weibull-Geometric distribution which generalizes the exponential-geometric distribution proposed by Adamidis and Loukas (1998). The hazard function of the last distribution is monotone decreasing but the hazard function of the new distribution can take more general forms. Unlike the Weibull distribution, the proposed distribution is useful for modeling unimodal failure rates. We derive the cumulative distribution and hazard functions, the density of the order statistics and calculate expressions for its moments and for the moments of the order statistics. We give expressions for the R\'enyi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an algorithm EM (Dempster et al., 1977; McLachlan and Krishnan, 1997) is provided for estimating the parameters. We obtain the information matrix and discuss inference. Applications to real data sets are given to show the flexibility and potentiality of the proposed distribution.