Theoretical analysis and improvements in cubic transmutations of probability distributions

TL;DR

This paper compares various cubic transmutation methods for extending probability distributions, analyzes their theoretical relationships, proposes modifications, and illustrates findings using Pareto distribution.

Contribution

It provides a comprehensive comparison of different cubic transmutation approaches and introduces modified versions with extended parameter ranges.

Findings

Different CT approaches are related and can be unified.

Modified CT methods improve flexibility of distribution modeling.

Empirical tests on Pareto distribution demonstrate enhanced performance.

Abstract

In statistics, processed data are becoming increasingly complex, and classical probability distributions are limited in their ability to model them. This is why, to better model data, extensive work has been conducted on extending classical probability distributions. Generally, this extension is achieved by transforming the cumulative distribution function of a baseline distribution through the addition of one or more parameters to enhance its flexibility. Cubic transmutation (CT) is one of the most popular methods for such extensions. However, CT does not have a unique definition because different approaches for CT have been proposed in the literature but are yet to be compared. The main goal of this paper is to compare these different approaches from both theoretical and empirical viewpoints. We study the relationships between the different approaches and we propose modified versions based on the extension of parameter ranges. The results are illustrated using Pareto distribution as baseline distribution.