A new discrete Pham model for nonmonotone aging and overdispersion: theory, inference, optimization, and three scientific applications

Ahmed Elshahhat; Osama E. Abo-Kasem; Heba S. Mohammed; Ahmed Elshahhat; Osama E. Abo-Kasem; Heba S. Mohammed

doi:10.3934/math.2026411

AIMS Mathematics

2026, Volume 11, Issue 4: 9942-9988. doi: 10.3934/math.2026411

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Research article Special Issues

A new discrete Pham model for nonmonotone aging and overdispersion: theory, inference, optimization, and three scientific applications

1.
Faculty of Technology and Development, Zagazig University, Zagazig 44519, Egypt
2.
Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig 44519, Egypt
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 26 January 2026 Revised: 24 March 2026 Accepted: 07 April 2026 Published: 13 April 2026
MSC : 60E05, 62E10, 62N01, 62N05, 62P10

Discrete lifetime data arising in engineering reliability, biomedical studies, and risk-based applications often display nonmonotone aging behavior, heavy tails, and substantial overdispersion, which challenge the adequacy of classical discrete models. To overcome these limitations, the paper proposes a new discrete Pham (DPham) distribution obtained through survival-function discretization of the continuous Pham model. This construction yields a parsimonious yet highly flexible two-parameter discrete lifetime family that substantially extends the modeling capacity of existing discrete distributions. A central contribution of the DPham model is its ability to accommodate a wide range of hazard rate shapes, including decreasing, increasing, constant, and bathtub forms, within a single, unified framework. This flexibility allows the model to capture complex aging mechanisms such as early-failure dominance, intermediate-life stabilization, and long-tailed persistence while retaining analytical tractability. The probability mass function supports both monotone and unimodal configurations, enhancing its descriptive power in discrete settings. The paper develops comprehensive distributional properties of the DPham distribution, including quantile representation, dispersion characteristics, skewness, kurtosis, and order statistics, demonstrating a level of adaptability that exceeds that of commonly used discrete lifetime models. Inferential procedures are systematically explored under both likelihood-based and Bayesian frameworks, with explicit consideration of Type-Ⅱ censored data. Bayesian estimation, implemented using Markov iterative algorithms, is shown to provide improved estimation accuracy and more reliable interval estimates, particularly in small-sample and heavily censored scenarios. The practical utility of the DPham distribution is illustrated through applications to real datasets from clinical survival analysis, industrial reliability systems, and overdispersed risk data. Across all cases, the proposed model consistently outperforms nine established discrete competitors, including discrete Weibull-type and gamma-based models, in terms of goodness of fit, inferential stability, and robustness to extreme observations. Supported by extensive simulation evidence, these results establish the DPham distribution as a versatile and domain-agnostic tool for discrete lifetime modeling, offering a compelling alternative for reliability theory, biostatistics, and applied risk analysis.
- Pham,
- Ⅴ-tub shaped,
- survival,
- hazard,
- Bayesian,
- likelihood,
- normality and log-normality,
- moments,
- order statistics,
- optimization,
- Markov chain,
- Newton-Raphson,
- censoring,
- data applicability
Citation: Ahmed Elshahhat, Osama E. Abo-Kasem, Heba S. Mohammed. A new discrete Pham model for nonmonotone aging and overdispersion: theory, inference, optimization, and three scientific applications[J]. AIMS Mathematics, 2026, 11(4): 9942-9988. doi: 10.3934/math.2026411

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Abstract

Discrete lifetime data arising in engineering reliability, biomedical studies, and risk-based applications often display nonmonotone aging behavior, heavy tails, and substantial overdispersion, which challenge the adequacy of classical discrete models. To overcome these limitations, the paper proposes a new discrete Pham (DPham) distribution obtained through survival-function discretization of the continuous Pham model. This construction yields a parsimonious yet highly flexible two-parameter discrete lifetime family that substantially extends the modeling capacity of existing discrete distributions. A central contribution of the DPham model is its ability to accommodate a wide range of hazard rate shapes, including decreasing, increasing, constant, and bathtub forms, within a single, unified framework. This flexibility allows the model to capture complex aging mechanisms such as early-failure dominance, intermediate-life stabilization, and long-tailed persistence while retaining analytical tractability. The probability mass function supports both monotone and unimodal configurations, enhancing its descriptive power in discrete settings. The paper develops comprehensive distributional properties of the DPham distribution, including quantile representation, dispersion characteristics, skewness, kurtosis, and order statistics, demonstrating a level of adaptability that exceeds that of commonly used discrete lifetime models. Inferential procedures are systematically explored under both likelihood-based and Bayesian frameworks, with explicit consideration of Type-Ⅱ censored data. Bayesian estimation, implemented using Markov iterative algorithms, is shown to provide improved estimation accuracy and more reliable interval estimates, particularly in small-sample and heavily censored scenarios. The practical utility of the DPham distribution is illustrated through applications to real datasets from clinical survival analysis, industrial reliability systems, and overdispersed risk data. Across all cases, the proposed model consistently outperforms nine established discrete competitors, including discrete Weibull-type and gamma-based models, in terms of goodness of fit, inferential stability, and robustness to extreme observations. Supported by extensive simulation evidence, these results establish the DPham distribution as a versatile and domain-agnostic tool for discrete lifetime modeling, offering a compelling alternative for reliability theory, biostatistics, and applied risk analysis.

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