Estimation of a new parameter discriminating between Weibull tail-distributions and heavy-tailed distributions
Résumé
The Gnedenko theorem is a general result in extreme value theory establishing the asymptotic distribution of extreme order statistics. The maximum of a sample of independent and identically distributed random variables after proper renormalization converges in distribution to one of the three possible maximum domain of attraction: Fr échet, Weibull and Gumbel. In a lot of applications (hydrology, nance, etc...), the Fr échet maximum domain of attraction and the Gumbel maximum domain of attraction are used. The Gumbel maximum domain of attraction encompasses a large variety of distributions. Here, we focus on a subfamily of distributions called Weibull tail-distributions which depends on the Weibull tail-coeffi cient. Numerous works are dedicated to the estimation of this coeffi cient, and to the estimation of the tail index. In order to explain why the same methodology can be used to estimate the Weibull tail-coeffi cient and the tail index, the authors proposed a family of distributions which encompasses the whole Fr échet maximum domain of attraction as well as Weibull tail-distributions. These distributions depend on 2 parameters $\tau\in[0,1]$ and $\theta>0$. The fi rst one, $\tau$, allows us to represent a large panel of distribution tails ranging from Weibull-type tails ($\tau=0$) to distributions belonging to the maximum domain of attraction of Fr échet ($\tau=1$). The main goal of this communication is to propose an estimator for $\tau$ independent of $\theta$. Under some assumptions we establish the asymptotic distribution of this estimator.