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

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.