A Generalization of the Exponential-Poisson Distribution

TL;DR

This paper introduces a generalized exponential-Poisson distribution with flexible failure rate shapes, providing comprehensive mathematical properties, estimation methods, and real data applications.

Contribution

It extends the exponential-Poisson distribution to allow various failure rate shapes and offers detailed mathematical, statistical, and practical analysis.

Findings

Distribution can have decreasing, increasing, or upside-down bathtub failure rates.

Closed-form expressions for density, distribution, and moments are derived.

Applications demonstrate the distribution's flexibility on real data.

Abstract

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the $i$th order statistic. We derive the $r$th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the R\'enyi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented.