The Poisson two-sum XLindley distribution: Statistical properties, estimation, regression modeling, and applications

Hanan Haj Ahmad; Mahmoud M. El-Awady; Hanan Haj Ahmad; Mahmoud M. El-Awady

doi:10.3934/math.20261122

AIMS Mathematics

2026, Volume 11, Issue 9: 28121-28160. doi: 10.3934/math.20261122

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Research article

The Poisson two-sum XLindley distribution: Statistical properties, estimation, regression modeling, and applications

Hanan Haj Ahmad ^{1
,
,},
Mahmoud M. El-Awady ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 31982, Al-Ahsa, Saudi Arabia
2.
Basic Sciences Department, Misr Higher Institute for Commerce and Computers, Mansoura, Egypt

Received: 05 July 2026 Revised: 20 August 2026 Accepted: 25 August 2026 Published: 03 September 2026
MSC : 60E05, 62E15, 62F10, 62J12

This paper introduces a new one-parameter count model, called the Poisson two-sum XLindley (PTS-XL) distribution, obtained by compounding the Poisson distribution with the two-sum XLindley distribution, defined as the sum of two independent XLindley random variables with a common parameter. The PTS-XL distribution provides a flexible framework for modeling overdispersed and positively skewed data. Several mathematical and statistical properties of the proposed models are derived. Different estimation procedures are developed to estimate the model's parameter, and the performance of the estimators is assessed through simulation studies. Furthermore, a count regression model based on the PTS-XL distribution is developed to model count response data in the presence of covariates. The regression parameters are estimated using the maximum likelihood estimate and assessed through a simulation study. The adequacy of the fitted model is assessed using residual analysis. Finally, real count datasets are analyzed to evaluate the applicability of the proposed distribution and regression model. The results show that the PTS-XL distribution provides a useful and flexible alternative for modeling overdispersed count data.
- mixed-Poisson distribution,
- XLindley distribution,
- two-sum XLindley distribution,
- maximum likelihood estimation,
- count regression,
- simulation study
Citation: Hanan Haj Ahmad, Mahmoud M. El-Awady. The Poisson two-sum XLindley distribution: Statistical properties, estimation, regression modeling, and applications[J]. AIMS Mathematics, 2026, 11(9): 28121-28160. doi: 10.3934/math.20261122

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Abstract

This paper introduces a new one-parameter count model, called the Poisson two-sum XLindley (PTS-XL) distribution, obtained by compounding the Poisson distribution with the two-sum XLindley distribution, defined as the sum of two independent XLindley random variables with a common parameter. The PTS-XL distribution provides a flexible framework for modeling overdispersed and positively skewed data. Several mathematical and statistical properties of the proposed models are derived. Different estimation procedures are developed to estimate the model's parameter, and the performance of the estimators is assessed through simulation studies. Furthermore, a count regression model based on the PTS-XL distribution is developed to model count response data in the presence of covariates. The regression parameters are estimated using the maximum likelihood estimate and assessed through a simulation study. The adequacy of the fitted model is assessed using residual analysis. Finally, real count datasets are analyzed to evaluate the applicability of the proposed distribution and regression model. The results show that the PTS-XL distribution provides a useful and flexible alternative for modeling overdispersed count data.

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