Mean Squared Error Representative Points of Pareto Distributions and Their Estimation

Area of Science:

• Statistics
• Probability Theory
• Data Modeling

Background:

• Pareto distributions are extensively used across economics, finance, and environmental science.
• Accurate discrete representations of Pareto distributions are essential for modeling real-world data.
• Existing methods may lack efficiency or accuracy in discretizing Pareto distributions.

Purpose of the Study:

• To propose and validate mean squared error representative points (MSE-RPs) as a discrete representation for Pareto distributions.
• To develop theoretical and computational methods for calculating MSE-RPs.
• To investigate and recommend optimal methods for estimating MSE-RPs, addressing estimation bias.

Main Methods:

• Development of a theoretical k-means algorithm for computing MSE-RPs for Pareto I and II distributions.
• Employment of three distinct methodological approaches for estimating MSE-RPs.
• Analysis of estimation bias across different parameters and methods, including information gain truncation for determining the number of MSE-RPs.

Main Results:

• Demonstrated the existence and uniqueness of MSE-RPs under specific parameter conditions.
• Recommended parameter estimation prior to MSE-RP estimation for Pareto I and II distributions.
• Suggested Bq quantiles for MSE-RP estimation in Pareto III and IV distributions.
• Validated proposed estimation methods through simulations and real-world data analysis, showing accurate fitting of empirical distribution functions.

Conclusions:

• MSE-RPs offer an effective discrete representation for Pareto distributions.
• The proposed estimation strategies, including parameter estimation first or using Bq quantiles, improve applicability.
• The developed methods accurately fit empirical data, enhancing the utility of discrete Pareto models.