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The beta Gumbel distribution. (English) Zbl 1068.62012
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
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