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.

摘要

本文研究的离散帕累托（DP）分布是一种具有幂律尾部的概率模型，它为众所周知的齐普夫分布提供了方便的替代方案。虽然 DP 模型的基本特征是明确可用的，但这不是一个指数族，并且与该模型相关的参数估计是一项具有挑战性的任务。考虑到这一点，我们开发了一种基于期望最大化（EM）算法的计算方法来解决这个问题。在此过程中，我们发现了一个有趣的新概率分布，它是标准伽马模型的某种倾斜版本，并且我们提供了对其基本属性的简短说明。后者在我们的 EM 算法中起着至关重要的作用。我们的 DP 参数估计计算方法通过模拟进行说明，