CLOSED-FORM EXPRESSIONS AND THE ASYMPTOTICS TO MOMENTS FOR THE EXCESS GOMPERTZ–MAKEHAM DISTRIBUTION

Gompertz's law, published 200 years ago, became the main demographic model. It postulated an exponential increase in human survival risk with age. Since then, this model and its extensions have been successfully used in biology, actuarial science, and other fields. These models can be used in applications related to reliability theory. This article derives an exact formula for Gompertz–Makeham residual life expectancy and for residual variance that can serve for computational convenience. The moments are expressed in a closed form by using a generalized integro-exponential function or Meyer's G function. This allows direct calculation using standard software. In addition, data analysis of complex systems shows that datasets can often be characterized by the behavior of probability distributions at great age. The asymptotic expansions of the first and second residual moments and the error estimate are obtained for large argument values. The results presented in the article can serve as a tool for applications in the theory of risk, reliability and extreme events.