New families of the Gumbel distribution: Bimodality, regression, and applications

Abstract

This paper proposes a new generalization of the univariate Gumbel distribution between theory and application. With only one scale parameter and one location parameter, the Gumbel distribution is known to model many real-world applications related to extreme weather conditions, engineering, and more. Because of its importance, statisticians are constantly looking to increase its flexibility and applicability by using it as a base distribution in many generalization methods. The first part of this paper presents a list of some of the most well-known recent Gumbel generalizations, based on different generalization techniques. In the second part, we employ theoretical principles to introduce a new generalization of the Gumbel distribution using the $T$-$R\{Y\}$ framework. In the application section, two real-life data sets are used to compare the goodness of fit of our new generalization to that of competitive distributions in modeling unimodal and bimodal data. Lastly, a regression model with censored data is presented for one of the members of the new generalization.

Keywords

• Gumbel distribution
• Gumbel generalizations
• T-R{Y} framework

Supporting Institution

The first author gratefully acknowledges the support received from the Department of Mathematics at the State University of New York at Farmingdale (SUNY-FSC) during the summer.

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