The beta Gumbel distribution. (English) Zbl 1068.62012

Nadarajah, Saralees; Kotz, Samuel

Math. Probl. Eng. 2004, No. 4, 323-332 (2004).

Summary: The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. 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 \(n\)-th 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.

MSC:

62E15	Exact distribution theory in statistics
62N99	Survival analysis and censored data
62P30	Applications of statistics in engineering and industry; control charts
62E20	Asymptotic distribution theory in statistics
33C90	Applications of hypergeometric functions
62E10	Characterization and structure theory of statistical distributions
62G32	Statistics of extreme values; tail inference