Mathematical Problems in Engineering
Volume 2004 (2004), Issue 4, Pages 323-332

The beta Gumbel distribution

Saralees Nadarajah1 and Samuel Kotz2

1Department of Mathematics, University of South Florida, Tampa 33620, FL, USA
2Department of Engineering Management and Systems Engineering, The George Washington University, Washington DC 20052, USA

Received 22 March 2004; Revised 16 June 2004

Copyright © 2004 Saralees Nadarajah and Samuel Kotz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.