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Article

Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events

by
Cristian Gabriel Anghel
Faculty of Hydrotechnics, Technical University of Civil Engineering Bucharest, Lacul Tei, Nr. 122–124, 020396 Bucharest, Romania
Mathematics 2024, 12(16), 2466; https://doi.org/10.3390/math12162466
Submission received: 11 July 2024 / Revised: 24 July 2024 / Accepted: 8 August 2024 / Published: 9 August 2024

Abstract

The manuscript presents the applicability of the Gumbel distribution in the frequency analysis of extreme events in hydrology. The advantages and disadvantages of using the distribution are highlighted, as well as recommendations regarding its proper use. A literature review was also carried out regarding the methods for estimating the parameters of the Gumbel distribution in hydrology. Thus, for the verification of the methods, case studies are presented regarding the determination of the maximum annual flows and precipitations using nine methods for estimating the distribution parameters. The influence of the variability of the observed data lengths on the estimation of the statistical indicators, the estimation of the parameters, and the quantiles corresponding to the field of small exceedance probabilities (p < 1%) is also highlighted. In each case, the results are analyzed compared to those obtained with the Generalized Extreme Value distribution, the four-parameter Burr distribution, and the five-parameter Wakeby distribution estimated using the L-moments method. The results of the case studies highlight and reaffirm the statistical, mathematical, and hydrological recommendations regarding the avoidance of applying the Gumbel distribution in flood frequency analysis and its use with reservations in the case of maximum precipitation analysis, especially when the statistical indicators of the analyzed data are not close to the characteristic ones and unique to the distribution.
Keywords: Gumbel; parameter estimation; variability; theoretical bias; frequency analysis Gumbel; parameter estimation; variability; theoretical bias; frequency analysis

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MDPI and ACS Style

Anghel, C.G. Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events. Mathematics 2024, 12, 2466. https://doi.org/10.3390/math12162466

AMA Style

Anghel CG. Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events. Mathematics. 2024; 12(16):2466. https://doi.org/10.3390/math12162466

Chicago/Turabian Style

Anghel, Cristian Gabriel. 2024. "Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events" Mathematics 12, no. 16: 2466. https://doi.org/10.3390/math12162466

APA Style

Anghel, C. G. (2024). Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events. Mathematics, 12(16), 2466. https://doi.org/10.3390/math12162466

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