Reconstruction of Frailty- based Mortality Models by a Generalisation of Gompertz Distribution

Heterogeneous mortality models are one of the most important subjects in survival analysis studies. In such models, because of individual and genetic differences between individuals which are influenced on their lifetime, we encounter heterogeneous populations. To make clear this heterogeneity, we use a non-negative real valued quantity called frailty. Since frailty variable is an unobservable quantity, so it is not possible to measure it. To consider this variable in mortality models by introducing a generalized Gompertz distribution, we show that in heterogeneous mortality models with Gompertz as a baseline force of mortality and Gamma distributed frailty, the lifetime random variable can be considered as a difference of two generalized Gompertz random variables. In this difference, the first random variable indicates the population's force of mortality which is distributed with Gompertz and the second random variable indicates frailty. Due to the relation between generalized Gompertz distribution and its reversed version, we consider the frailty variable with reversed generalized Gompertz distribution. The method used in this article allows the difference distribution to interpret the frailty distribution as a lifetime reduction distribution.