A flexible two-parameter discrete distribution with a bathtub-shaped hazard function: statistical properties and applications

Ammar M. Sarhan; Najwan Alsadat; M. E. Sobh; Ammar M. Sarhan; Najwan Alsadat; M. E. Sobh

doi:10.3934/math.20261292

AIMS Mathematics

2026, Volume 11, Issue 10: 32945-32972. doi: 10.3934/math.20261292

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Research article

A flexible two-parameter discrete distribution with a bathtub-shaped hazard function: statistical properties and applications

1.
Department of Mathematics and Statistics, Dalhousie University, Nova Scotia, Canada
2.
Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received: 09 August 2026 Revised: 20 September 2026 Accepted: 21 September 2026 Published: 10 October 2026
MSC : Primary 62G32, Secondary 60F05, 62G09, 62G20, 62G30

This work introduced a new two-parameter discrete distribution derived from a continuous truncated lifetime model. The proposed distribution features a flexible hazard rate function that can exhibit both increasing and bathtub-shaped behaviors depending on parameter configurations, allowing it to capture diverse patterns in discrete failure time data. Despite its flexibility, the model retains a parsimonious two-parameter structure, facilitating straightforward statistical inference. We derived key structural and statistical properties of the model and investigated parameter estimation under both maximum likelihood and Bayesian frameworks. Because closed-form expressions for the estimators are analytically intractable, numerical methods were developed; specifically, maximum likelihood estimates were obtained via numerical optimization in R, while Bayesian estimates were computed using a Markov chain Monte Carlo sampler. The finite-sample performance of both estimation procedures was evaluated through an extensive Monte Carlo simulation study. Finally, the practical utility of the distribution was demonstrated through applications to four real-world datasets across different fields, where it yielded competitive fits relative to several established discrete lifetime models.
- discrete distribution,
- bathtub-shaped hazard rate,
- discretization,
- maximum likelihood estimation,
- Bayesian estimation
Citation: Ammar M. Sarhan, Najwan Alsadat, M. E. Sobh. A flexible two-parameter discrete distribution with a bathtub-shaped hazard function: statistical properties and applications[J]. AIMS Mathematics, 2026, 11(10): 32945-32972. doi: 10.3934/math.20261292

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Abstract

This work introduced a new two-parameter discrete distribution derived from a continuous truncated lifetime model. The proposed distribution features a flexible hazard rate function that can exhibit both increasing and bathtub-shaped behaviors depending on parameter configurations, allowing it to capture diverse patterns in discrete failure time data. Despite its flexibility, the model retains a parsimonious two-parameter structure, facilitating straightforward statistical inference. We derived key structural and statistical properties of the model and investigated parameter estimation under both maximum likelihood and Bayesian frameworks. Because closed-form expressions for the estimators are analytically intractable, numerical methods were developed; specifically, maximum likelihood estimates were obtained via numerical optimization in R, while Bayesian estimates were computed using a Markov chain Monte Carlo sampler. The finite-sample performance of both estimation procedures was evaluated through an extensive Monte Carlo simulation study. Finally, the practical utility of the distribution was demonstrated through applications to four real-world datasets across different fields, where it yielded competitive fits relative to several established discrete lifetime models.

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