On order statistics from the Gompertz–Makeham distribution and the Lambert W function

Abstract

The aim of this paper is twofold. First, we show that the expected value of the minimum order statistic from the Gompertz-Makeham distribution can be expressed in closed form in terms of the incomplete gamma function. We also give a general formula for the moments of the minimum order statistic in terms of the generalized integro-exponential function. As a consequence, the moments of all order statistics from this probability distribution can be more easily evaluated from the moments of the minimum order statistic. Second, we show that the maximum and minimum order statistics from the Gompertz-Makeham distribution are in the domains of attraction of the Gumbel and Weibull distributions, respectively. Lambert W function plays an important role in solving these problems.