The heavy-tailed chi-square model: properties, estimation and application to wind speed data

Eliseo Martínez; Emilio Gómez-Déniz; Diego I. Gallardo; Osvaldo Venegas; Héctor W. Gómez; Eliseo Martínez; Emilio Gómez-Déniz; Diego I. Gallardo; Osvaldo Venegas; Héctor W. Gómez

doi:10.3934/math.20251060

AIMS Mathematics

2025, Volume 10, Issue 10: 23849-23868. doi: 10.3934/math.20251060

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The heavy-tailed chi-square model: properties, estimation and application to wind speed data

1.
Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile
2.
Department of Quantitative Methods in Economics and TIDES Institute, University of Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
3.
Departamento de Estadística, Facultad de Ciencias, Universidad del Bío-Bío, Concepción 4081112, Chile
4.
Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingeniería, Universidad Católica de Temuco, Temuco 4780000, Chile
5.
Departamento de Estadística y Ciencia de Datos, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile

Received: 06 September 2025 Revised: 06 October 2025 Accepted: 13 October 2025 Published: 21 October 2025
MSC : 62E15, 62E20, 62F10, 62P99

In this article, we introduced an extension of the chi-square distribution by employing a slash-type methodology that enhanced the weight of the right tail, thereby producing a heavy-tailed distribution. We explored two different representations of the proposed distribution and examined several of its key properties, such as the mode, cumulative distribution function, reliability and hazard functions, moments, and the skewness and kurtosis coefficients. Additionally, we demonstrated that the classical chi-square distribution was a special case of our proposed model. Parameter estimation was carried out using both the method of moments and the maximum likelihood estimation, the latter via the expectation-maximization (EM) algorithm. A simulation study was conducted to evaluate the performance of parameter recovery. Finally, we applied the new distribution to a wind speed dataset, showing that it provided a good fit, particularly in the presence of extreme values.
- chi-square distribution,
- EM algorithm,
- heavy-tailed distribution,
- maximum likelihood,
- slash distribution
Citation: Eliseo Martínez, Emilio Gómez-Déniz, Diego I. Gallardo, Osvaldo Venegas, Héctor W. Gómez. The heavy-tailed chi-square model: properties, estimation and application to wind speed data[J]. AIMS Mathematics, 2025, 10(10): 23849-23868. doi: 10.3934/math.20251060

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Abstract

In this article, we introduced an extension of the chi-square distribution by employing a slash-type methodology that enhanced the weight of the right tail, thereby producing a heavy-tailed distribution. We explored two different representations of the proposed distribution and examined several of its key properties, such as the mode, cumulative distribution function, reliability and hazard functions, moments, and the skewness and kurtosis coefficients. Additionally, we demonstrated that the classical chi-square distribution was a special case of our proposed model. Parameter estimation was carried out using both the method of moments and the maximum likelihood estimation, the latter via the expectation-maximization (EM) algorithm. A simulation study was conducted to evaluate the performance of parameter recovery. Finally, we applied the new distribution to a wind speed dataset, showing that it provided a good fit, particularly in the presence of extreme values.

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