From exponential-geometric to Lomax: A unified survival model via gamma frailty

Mohieddine Rahmouni; Mohieddine Rahmouni

doi:10.3934/math.20251315

AIMS Mathematics

2025, Volume 10, Issue 12: 29927-29954. doi: 10.3934/math.20251315

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From exponential-geometric to Lomax: A unified survival model via gamma frailty

Mohieddine Rahmouni ^,

Applied College, King Faisal University, Al-Ahsa, Saudi Arabia

Received: 07 September 2025 Revised: 22 October 2025 Accepted: 11 November 2025 Published: 19 December 2025
MSC : 62E15, 60E05, 62F10

This paper introduces a new flexible lifetime distribution that unifies the exponential-geometric and Lomax models through a random system size and a shared gamma frailty. It models the first failure time in a system with a geometrically distributed number of conditionally exponential components with gamma frailty, producing a geometric mixture of Lomax distributions. The model generalizes classical distributions, reducing to exponential-geometric, Lomax, or exponential under appropriate limits. We derived analytical expressions for the cumulative distribution, survival, and hazard functions, discussed maximum likelihood estimation, and illustrated its competitive and versatile fits on real datasets.
- survival analysis,
- gamma frailty,
- geometric distribution,
- Lomax distribution,
- mixture model,
- reliability,
- first failure time
Citation: Mohieddine Rahmouni. From exponential-geometric to Lomax: A unified survival model via gamma frailty[J]. AIMS Mathematics, 2025, 10(12): 29927-29954. doi: 10.3934/math.20251315

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Abstract

This paper introduces a new flexible lifetime distribution that unifies the exponential-geometric and Lomax models through a random system size and a shared gamma frailty. It models the first failure time in a system with a geometrically distributed number of conditionally exponential components with gamma frailty, producing a geometric mixture of Lomax distributions. The model generalizes classical distributions, reducing to exponential-geometric, Lomax, or exponential under appropriate limits. We derived analytical expressions for the cumulative distribution, survival, and hazard functions, discussed maximum likelihood estimation, and illustrated its competitive and versatile fits on real datasets.

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