The beta Gumbel distribution

By Saralees Nadarajah, Samuel Kotz. 2004

293 citations recorded by the source corpus.

The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. In this paper, we introduce a generalization—referred to as the beta Gumbel distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of this new distribution. We derive the analytical shapes of the corresponding probability density function and the hazard rate function and provide graphical illustrations. We calculate expressions for the nth moment and the asymptotic distribution of the extreme order statistics. We investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. We hope that this generalization will attract wider applicability in engineering.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3cf3c442ec3099e00100eb21dd0f7f163846c3da

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Citations in the corpus

The beta exponentiated Weibull distribution (2013, background): “Eugene et al. [9], Nadarajah and Kotz [11], Nadarajah and Gupta [12], Nadarajah and Kotz [2] and Barreto-Souza et al. [13] proposed the beta normal, beta Gumbel, beta Fréchet, beta exponential (BE) and beta exponentiated exponential (BEE) distributions by taking G(x) in Equation (2) to be the cdf of”
Topp–Leone Family of Distributions: Some Properties and Application (2016, unknown): “The Beta family of distribution has been widely studied for different choices of G x , see for example Famoye et al. (2005), Hanook et al. (2013), Nadarajah and Kotz (2004) among many others.”
The new class of Kummer beta generalized distributions: theory and applications (2013, unknown): “Following Eugene et al. (2002), who defined the beta normal (BN) distribution, Nadarajah and Kotz (2004) introduced the beta Gumbel distribution (BGu), provided expressions for the moments, examined the asymptotic distribution of the extreme order statistics and performed maximum likelihood…”
A New Class of Generalized Power Lindley Distribution with Applications to Lifetime Data (2016, unknown): “Nadarajah and Kotz [13], Nadarajah and Gupta [14], Nadarajah and Kotz [15], Barreto-Souza et al.”
A new class of generalized Lindley distributions with applications (2015, background): “[8] Nadarajah and Kotz,[9] Nadarajah and Gupta,[10] Nadarajah and Kotz,[11] and Barreto-Souza et al.”
The beta generalized gamma distribution (2013, background): “[14], Nadarajah and Kotz [17], Nadarajah and Gupta [18] and Nadarajah and Kotz [19] defined the beta normal, beta Gumbel, beta Fréchet and beta exponential distributions by taking G(x) to be the cdf of the normal, Gumbel, Fréchet and exponential distributions, respectively.”
The beta-Pareto distribution (2008, background): “Various authors have studied the density in Equation (3) by assuming various cumulative distribution functions F(x) (see [16, 18–20]).”
A new method for generating families of continuous distributions (2013, methodology): “Since the paper by Eugene et al. [10], many beta-generated distributions have been studied in the literature including the beta-Gumbel distribution by Nadarajah and Kotz [29], betaexponential distribution by Nadarajah and Kotz [30], beta-Weibull distribution by Famoye et al. [12], beta-gamma by dist”
THE BETA GENERALIZED PARETO DISTRIBUTION (2011, result): “Later, attempts- by Nadarajah and Gupta [11], Nadarajah and Kotz [10, 14], Kong et al.”
Closed-form expressions for moments of a class of beta generalized distributions (2011, unknown): “Nadarajah and Kotz (2004) introduced the beta Gumbel distribution by taking G(x) to be the cdf of the Gumbel distribution and provided closed form expressions for the moments and discussed the asymptotic distribution of the extreme order statistics and the maximum likelihood estimation procedure.”
The Beta-Dagum Distribution: Definition and Properties (2013, result): “Furthermore, we compare the obtained results with analogous ones obtained for the Da , the Beta Burr XII (or Beta Singh-Maddala (BSM)) (proposed by Paranaíba et al. (2011)) and the Burr XII (Singh-Maddala (SM)).”
Exponentiated $T$-$X$ Family of Distributions with Some Applications (2013, methodology): “Some examples include the beta Gumbel distribution by Nadarajah and Kotz (2004), the beta Fréchet distribution by Nadarajah and Gupta (2004), the beta-Weibull distribution by Famoye et al. (2005), the beta exponential distribution by Nadarajah and Kotz (2006), the beta-gamma distribution by Kong…”
Some Results for Beta Fréchet Distribution (2008, background): “Nadarajah and Kotz (2004) also introduced the beta Gumbel (BG) distribution by taking G(x) to be the cdf of the Gumbel distribution and provided closed-form expressions for the moments, the asymptotic distribution of the extreme order statistics and discussed the maximum likelihood estimation…”