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

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.