A Generalization of the Exponential-Poisson Distribution

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.

💡 Research Summary

The paper introduces a three‑parameter extension of the exponential‑Poisson (EP) distribution, originally proposed by Kus (2007), to overcome the limitation that the classic EP distribution can only exhibit a decreasing failure (hazard) rate. By adding a shape parameter θ to the existing scale (β) and rate (α) parameters, the authors define the Generalized Exponential‑Poisson (GEP) distribution with probability density

 f(t;α,β,θ)=α β (1+θ t) exp