The Weibull-generalized shifted geometric distribution: properties, estimation, and applications

Mohieddine Rahmouni; Dalia Ziedan; Mohieddine Rahmouni; Dalia Ziedan

doi:10.3934/math.2025448

AIMS Mathematics

2025, Volume 10, Issue 4: 9773-9804. doi: 10.3934/math.2025448

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The Weibull-generalized shifted geometric distribution: properties, estimation, and applications

Mohieddine Rahmouni ^{1
,
,},
Dalia Ziedan ²

1.
Applied College, King Faisal University, Al-Ahsa, Saudi Arabia
2.
Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt

Received: 09 February 2025 Revised: 13 April 2025 Accepted: 22 April 2025 Published: 27 April 2025
MSC : 60E05, 62E15, 62F10

This paper introduces the Weibull-generalized shifted geometric (WGSG) distribution, a novel lifetime model integrating the Weibull and shifted geometric distributions to address complex lifetime data patterns. Extending the Weibull-geometric framework, this distribution models system reliability by focusing on the $ k $-th smallest lifetime—when $ k $ components fail—rather than the minimum. Key properties, including the probability density function, cumulative distribution function, and moments, were derived. Parameters were estimated using maximum likelihood, expectation-maximization, method of moments, and Bayesian approaches, with a simulation study comparing their performance. Applications to two real-world lifetime and reliability datasets demonstrated the distribution's superiority over classical models in handling challenging survival and reliability scenarios. This flexible model enhances the ability to capture diverse hazard behaviors, advancing lifetime data analysis.
- lifetime distributions,
- reliability,
- failure rate,
- order statistics,
- WGSG distribution,
- geometric distribution,
- entropy
Citation: Mohieddine Rahmouni, Dalia Ziedan. The Weibull-generalized shifted geometric distribution: properties, estimation, and applications[J]. AIMS Mathematics, 2025, 10(4): 9773-9804. doi: 10.3934/math.2025448

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Abstract

This paper introduces the Weibull-generalized shifted geometric (WGSG) distribution, a novel lifetime model integrating the Weibull and shifted geometric distributions to address complex lifetime data patterns. Extending the Weibull-geometric framework, this distribution models system reliability by focusing on the $ k $-th smallest lifetime—when $ k $ components fail—rather than the minimum. Key properties, including the probability density function, cumulative distribution function, and moments, were derived. Parameters were estimated using maximum likelihood, expectation-maximization, method of moments, and Bayesian approaches, with a simulation study comparing their performance. Applications to two real-world lifetime and reliability datasets demonstrated the distribution's superiority over classical models in handling challenging survival and reliability scenarios. This flexible model enhances the ability to capture diverse hazard behaviors, advancing lifetime data analysis.

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