A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

Abstract

The shifted Gompertz distribution was introduced by Bemmaor (1994) as a random model of adoption timing of innovations in a market. The purpose of this article is threefold. We provide explicit expressions for the expectation and variance of this model, which are functions of the Euler constant. In addition, for simulation purposes, we derive a closed-form expression for the quantile function of the shifted Gompertz distribution in terms of the Lambert W function. Finally, the limit distributions and computer generation of extreme order statistics are considered. In particular, we show that the domains of attraction for maxima and minima are the Gumbel and Weibull distributions, respectively.

Keywords:

• Extreme order statistics
• Moments
• Shifted Gompertz distribution
• Simulation

Mathematics Subject Classification:

• Primary 60E05
• Secondary 65C05

Acknowledgments

The authors wish to express their sincere thanks to the editors and anonymous referees for their careful reading of the manuscript and their valuables comments and suggestions, which led to an improvement of the article. This work was partially supported by research project MTM2005-08376-C02-01.