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The beta Fréchet distribution. (English) Zbl 1074.62008
Summary: Motivated by recent work of N. Eugene et al. [Commun. Stat., Theory Methods 31, No. 4, 497–512 (2002; Zbl 1009.62516)] and A. K. Gupta and S. Nadarajah [ibid. 33, No. 1, 1–13 (2004; Zbl 1210.60018)], we introduce the beta Fréchet 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.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62E20 Asymptotic distribution theory in statistics 62G32 Statistics of extreme values; tail inference 62N05 Reliability and life testing
##### Keywords:
hazard rate function