Maxima of Weibull-like distributions and the Lambert W function

Abstract: The Weibull--like distributions form a large class of probability distributions that belong to the domain of attraction for the maxima of the Gumbel law. Besides the Weibull distribution, it includes important distributions as the Gamma laws and, in particular, the $\chi^2$ distributions. In order to have explicit expressions of the norming constants for the maxima it is necessary to solve asymptotically a nonlinear equation; however, for some members of that family, numerical and simulation studies show that the constants that are usual suggested are inaccurate for moderate or even large sample sizes. In this paper we propose other norming constants computed with the asymptotics of the Lambert W function that significantly improve the accuracy of the approximation to the Gumbel law. These results are applied to the computation of the constants for the maxima of Gamma random variables that appear in some applied problems.