Modeling Non-Normal Distributions with Mixed Third-Order Polynomials of Standard Normal and Logistic Variables

Mohan D. Pant, Aditya Chakraborty () and Ismail El Moudden
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Mohan D. Pant: Department of Epidemiology, Biostatistics & Environmental Health, Joint School of Public Health, Macon & Joan Brock Virginia Health Sciences at Old Dominion University, Norfolk, VA 23529, USA
Aditya Chakraborty: Department of Epidemiology, Biostatistics & Environmental Health, Joint School of Public Health, Macon & Joan Brock Virginia Health Sciences at Old Dominion University, Norfolk, VA 23529, USA
Ismail El Moudden: Research and Infrastructure Service Enterprise, Macon & Joan Brock Virginia Health Sciences at Old Dominion University, Norfolk, VA 23529, USA

Mathematics, 2025, vol. 13, issue 6, 1-24

Abstract: Continuous data associated with many real-world events often exhibit non-normal characteristics, which contribute to the difficulty of accurately modeling such data with statistical procedures that rely on normality assumptions. Traditional statistical procedures often fail to accurately model non-normal distributions that are often observed in real-world data. This paper introduces a novel modeling approach using mixed third-order polynomials, which significantly enhances accuracy and flexibility in statistical modeling. The main objective of this study is divided into three parts: The first part is to introduce two new non-normal probability distributions by mixing standard normal and logistic variables using a piecewise function of third-order polynomials. The second part is to demonstrate a methodology that can characterize these two distributions through the method of L -moments (Mo L Ms) and method of moments (MoMs). The third part is to compare the Mo L Ms- and MoMs-based characterizations of these two distributions in the context of parameter estimation and modeling non-normal real-world data. The simulation results indicate that the Mo L Ms-based estimates of L -skewness and L -kurtosis are superior to their MoMs-based counterparts of skewness and kurtosis, especially for distributions with large departures from normality. The modeling (or data fitting) results also indicate that the Mo L Ms-based fits of these distributions to real-world data are superior to their corresponding MoMs-based counterparts.

Keywords: non-normal distribution; conventional moments; L -moments; mixed polynomials; Monte Carlo simulation; parameter estimation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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