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

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 (MoLMs) and method of moments (MoMs). The third part is to compare the MoLMs- 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 MoLMs-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 MoLMs-based fits of these distributions to real-world data are superior to their corresponding MoMs-based counterparts.