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

Abstract

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.

Keywords:

• moment-generating function
• expected value
• variance
• skewness
• kurtosis
• parameter estimation

Acknowledgements

The author sincerely thanks the editor and reviewer for their careful reading of the paper. The comments of the reviewer led to a substantial improvement of the manuscript. The author also wishes to thank Trifon I. Missov for discussions about the Gompertz distribution.

Notes

¹The Human Mortality Database uses a non-parametric estimate of exposures (Wilmoth et al. (2007)) and therefore it is assumed to be known.

²However, Pollard and Valkovics (1997) showed that an approximate likelihood for discrete data can be found using the truncated Gompertz distribution.

³For an alternative derivation (and also for alternative power series expansion of ) see the Appendix of Gussmann (1967), especially equations (A.39)−(A.41).

⁴S will be an s×n (Jacobian) matrix.