Reliability analysis for an independent Gompertz competing risks model employing improved adaptive progressive Type-Ⅱ censored data

Hana N. Alqifari; Hana N. Alqifari

doi:10.3934/math.20261043

AIMS Mathematics

2026, Volume 11, Issue 8: 26026-26076. doi: 10.3934/math.20261043

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Reliability analysis for an independent Gompertz competing risks model employing improved adaptive progressive Type-Ⅱ censored data

Hana N. Alqifari ^,

Department of Statistics and Operations Research, College of Science, Qassim University, Saudi Arabia

Received: 09 June 2026 Revised: 18 July 2026 Accepted: 29 July 2026 Published: 20 August 2026
MSC : 62N05, 62F15, 62F10

In engineering life tests, units frequently fail from several competing causes, such as electrical breakdown competing with surface erosion in an insulation system, while cost and time constraints require censored designs that guarantee both a fixed number of observed failures and a bounded test duration. Statistical inference for competing risks' data under such adaptive censoring schemes remains challenging because the censoring mechanism and latent failure causes jointly complicate parameter estimation. The improved adaptive Type Ⅱ progressive censoring scheme (IAT-Ⅱ PCS) satisfies both design requirements simultaneously, yet, to the best of our knowledge, no peer-reviewed inferential framework has been developed for Gompertz competing risks data under this scheme, despite the flexibility of the Gompertz distribution in modeling exponentially increasing hazard rates, which differ from the power-law hazard of the Weibull distribution and cannot be captured by the constant hazard of the exponential model. This paper develops unified classical and Bayesian inference for this model. Exploiting the log-linear structure of the Gompertz likelihood, closed-form maximum likelihood estimators are derived for the cause-specific scale parameters, reducing the optimization to a one-dimensional profile search, and exact Gamma full conditional posteriors are obtained, enabling an efficient hybrid Gibbs–Metropolis–Hastings sampler. A Monte Carlo simulation study shows that Bayesian estimators under moderately informative priors generally outperform maximum likelihood in finite samples, with substantially improved mixing for the scale parameters. Application to an electrode voltage–endurance dataset demonstrates that the Gompertz model provides a superior joint likelihood fit relative to the Weibull alternative, confirming its practical value for censored reliability analysis.
- Gompertz distribution,
- competing risks,
- improved adaptive Type Ⅱ progressive censoring,
- maximum likelihood estimation,
- hybrid MCMC,
- reliability analysis,
- simulation
Citation: Hana N. Alqifari. Reliability analysis for an independent Gompertz competing risks model employing improved adaptive progressive Type-Ⅱ censored data[J]. AIMS Mathematics, 2026, 11(8): 26026-26076. doi: 10.3934/math.20261043

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Abstract

In engineering life tests, units frequently fail from several competing causes, such as electrical breakdown competing with surface erosion in an insulation system, while cost and time constraints require censored designs that guarantee both a fixed number of observed failures and a bounded test duration. Statistical inference for competing risks' data under such adaptive censoring schemes remains challenging because the censoring mechanism and latent failure causes jointly complicate parameter estimation. The improved adaptive Type Ⅱ progressive censoring scheme (IAT-Ⅱ PCS) satisfies both design requirements simultaneously, yet, to the best of our knowledge, no peer-reviewed inferential framework has been developed for Gompertz competing risks data under this scheme, despite the flexibility of the Gompertz distribution in modeling exponentially increasing hazard rates, which differ from the power-law hazard of the Weibull distribution and cannot be captured by the constant hazard of the exponential model. This paper develops unified classical and Bayesian inference for this model. Exploiting the log-linear structure of the Gompertz likelihood, closed-form maximum likelihood estimators are derived for the cause-specific scale parameters, reducing the optimization to a one-dimensional profile search, and exact Gamma full conditional posteriors are obtained, enabling an efficient hybrid Gibbs–Metropolis–Hastings sampler. A Monte Carlo simulation study shows that Bayesian estimators under moderately informative priors generally outperform maximum likelihood in finite samples, with substantially improved mixing for the scale parameters. Application to an electrode voltage–endurance dataset demonstrates that the Gompertz model provides a superior joint likelihood fit relative to the Weibull alternative, confirming its practical value for censored reliability analysis.

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