Article/Chapter Title :
Statistics for Fréchet Distribution
Statistical Theory of Extremes
Fréchet distribution of maxima , Location and dispersion parameters , Weibull distribution of minima , Moment generating function , Estimation, testing, and point prediction , MLE , Confidence region
The Fréchet distribution for maxima is studied. Convenient transformations, leading to standard exponential, Gumbel and Weibull of minima, are given. From ML equations for dispersion and shape parameters, asymptotic normality is presented. Close to this, the asymptotic confidence region for the pair is obtained and similar results are presented for the quantile. The characteristics of the Fréchet distribution, moments and moment generating function are mentioned. Statistical decision for the 3-parameters case, estimation, testing, and point prediction are considered. In particular, MLE for location, dispersion and shape are given, solving ML equations, with first approximations using a quick tool from Gumbel. The overpassing probability for a level and a worked example close this section.
As regards moments, they exist only if r < α.
The asymptotic variance exists only if α >2.
The techniques for the Fréchet distribution of maxima and Weibull distribution of minima can be exchanged.
As far as applications are concerned, as said before, they have been done for largest waves and floods of rivers.