$\alpha$.

The most classical case corresponds to the setting where ${x}_{1-\alpha}$ is drawn from a $n$ sample of observations $X_1,\dots X_n$. We can distinguish estimation of high quantile, where $x_{1-\alpha}$ lies inside the sample range, see Weissman 1978 [@Weissman] and Dekkers and al. 1989 [@dekkers1989], and the estimation of an extreme quantile outside the boundary of the sample, see for instance De Haan and Rootzén 1993 [@deHann1993]. It is assumed that $X$ belongs to the domain of attraction of an extreme value distribution. The tail index of the latter is then estimated through maximum likelihood (Weissman 1978 [@Weissman]) or through an extension of Hill’s estimator (see the moment estimator by Dekkers and al. 1989 [@dekkers1989]). Lastly, the estimator of the quantile is deduced from the inverse function of the distribution of the $k$ largest observations. Note that all the above references assume that the distribution has a Pareto tail. An alternative modeling has been proposed by De Valk 2016 [@valk] and De Valk and Cai 2018 [@valk2], and consists in assuming a Weibull type tail, which enables to release some second order hypotheses on the tail. This last work deals with the estimation of extreme quantile lying way outside the 