The moments of the Gompertz distribution and maximum likelihood estimation of its parameters

The Gompertz distribution is widely used to describe the distribution of adult deaths. Previous works concentrated on formulating approximate relationships to characterise it. However, using the generalised integro-exponential function, exact formulas can be derived for its moment-generating function and central moments. Based on the exact central moments, higher accuracy approximations can be defined for them. In demographic or actuarial applications, maximum likelihood estimation is often used to determine the parameters of the Gompertz distribution. By solving the maximum likelihood estimates analytically, the dimension of the optimisation problem can be reduced to one both in the case of discrete and continuous data. Monte Carlo experiments show that by ML estimation, higher accuracy estimates can be acquired than by the method of moments.