Mean Squared Error Representative Points of Pareto Distributions and Their Estimation.

Abstract

Pareto distributions are widely applied in various fields, such as economics, finance, and environmental studies. The modeling of real-world data has created a demand for the discretization of Pareto distributions. In this paper, we propose using mean squared error representative points (MSE-RPs) as the discrete representation of Pareto distributions. We demonstrate the uniqueness and existence of these representative points under certain parameter settings and provide a theoretical k-means algorithm for the computation of MSE-RPs for Pareto I and Pareto II distributions. Furthermore, to enhance the applicability of MSE-RPs, we employ three methodological approaches to estimate the MSE-RPs of Pareto distributions. By analyzing the estimation bias under different parameters and methods, we recommend estimating the distribution parameters first before estimating the MSE-RPS for Pareto I and Pareto II distributions. For Pareto III and Pareto IV distributions, we suggest using the Bq quantiles for MSE-RP estimation. Building on this, we analyze the sources of estimation bias and propose an effective method for determining the number of MSE-RPs based on information gain truncation. Through simulations and real data studies, we demonstrate that the proposed methods for MSE-RP estimation are effective and can be used to fit the empirical distribution function of data accurately.