A computational approach to estimation of discrete Pareto parameters

Abstract

The discrete Pareto (DP) distribution studied in this paper is a probability model with a power-law tail, which provides a convenient alternative to the well-known Zipf distribution. While basic characteristics of the DP model are available explicitly, this is not an exponential family and parameter estimation connected with this model is a challenging task. With this in mind we develop a computational approach to this problem, based on the expectation-maximization (EM) algorithm. In the process, we discover an interesting new probability distribution, which is a certain tilted version of the standard gamma model, and we provide a short account of its basic properties. The latter play a crucial role in our EM algorithm. Our computational approach to DP parameter estimation is illustrated by simulations, while a real data example from finance illustrates potential applications of the DP stochastic model.

Keywords:

• Discrete distribution
• Discretization
• EM algorithm
• Heavy tails
• Parameter estimation
• Pareto distribution
• Power law
• Tilted distribution

MATHEMATICS SUBJECT CLASSIFICATION:

• 60E05
• 62E10
• 62E15
• 62F10
• 62P05

Acknowledgements

We thank the reviewers for their constructive comments that help improve the paper.