Sum of Independent XGamma Distributions

TL;DR

This paper studies the sum of independent XGamma distributions with different scale parameters, proving that the resulting PDF can be written as a finite linear combination of Erlang densities, which yields exact expressions for the CDF, MGF, and moments. It provides explicit coefficient forms via a Heaviside expansion and develops the HypoXG framework for modeling sums of XGamma variables, including closed-form functions for reliability and hazard. The authors then address parameter estimation through maximum likelihood, deriving implicit score equations and solving them numerically, illustrated with a bearing-life data set where HypoXG provides a competitive fit to Hypoexponential models. The work contributes a flexible, analytically tractable approach for reliability and survival analyses, offering closed-form related functions and a practical estimation strategy for mixture-of-exponential-type sums.

Abstract

The XGamma distribution is a generated distribution from a mixture of Exponential and Gamma distributions. It is found that in many cases the XGamma has more flexibility than the Exponential distribution. In this paper we consider the sum of independent XGamma distributions with different parameters. We showed that the probability density function of this distribution is a sum of the probability density function of the Erlang distributions. As a consequence, we find exact closed expressions of the other related statistical functions. Next, we examine the estimation of the parameters by maximum likelihood estimators. We observe in an applications a real data set which shows that this model provides better fit to the data as compared to the sum of the Exponential distributions, the Hypoexponential models.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 3.1