ORCID
0000-0002-8991-1737 (Pant), 0000-0002-1476-113X (Chakraborty), 0000-0001-7702-2564 (Moudden)
Document Type
Article
Publication Date
2025
DOI
10.3390/math13061019
Publication Title
Mathematics
Volume
13
Issue
6
Pages
1019 (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 (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.
Rights
© 2025 by the authors.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) License.
Data Availability
Article states: "For the data-fitting examples, simulated and publicly available datasets were used. The links to these data repositories are provided in the section where each dataset was mentioned. These links are provided again in the list below: (a) The random sample of n = 500 data from t-distribution with three degrees of freedom, used in Figure 4, was obtained using the Mathematica (Version 13.2) code SeedRandom [625]; (b) Data = RandomVariate[StudentTDistribution [3], 500]; (c) The random sample of n = 500 data from the stable distribution, used in Figure 5, was obtained using the Mathematica (Version 13.2) code SeedRandom [8924]; Data = RandomVariate[StableDistribution [1, 1.7, −0.4, 0, 1], 500]; (d) The daily return rate (percent change) of Apple stock (AAPL) for the 10 years between 18 August 2014 and 16 August 2024, used in Figure 6, was downloaded from the following website: https://www.nasdaq.com/symbol/aapl/historical (accessed on 18 August 2024). (e) The daily return rate (percent change) of Amazon stock (AMZN) for the 10 years between 18 August 2014 and 16 August 2024, used in Figure 7, was downloaded from the following website: (https://www.nasdaq.com/symbol/amzn/historical (accessed on 18 August 2024))."
Original Publication Citation
Pant, M. D., Chakraborty, A., & Moudden, I. E. (2025). Modeling non-normal distributions with mixed third-order polynomials of standard normal and logistic variables. Mathematics, 13(6), 1-24, Article 1019. https://doi.org/10.3390/math13061019
Repository Citation
Pant, M. D., Chakraborty, A., & Moudden, I. E. (2025). Modeling non-normal distributions with mixed third-order polynomials of standard normal and logistic variables. Mathematics, 13(6), 1-24, Article 1019. https://doi.org/10.3390/math13061019