To address this issue, we will consider a design of experiment enabling to progressively characterize the tail of the distribution by sampling at each step in a more extreme region of the density. It will thus be assumed in the following that we are able to observe $Y$ not only when $R$ follows $G$ but also when $R$ follows the conditional distribution of $R$ given $\{ R>x\}.$ In such a case we will be able to estimate $q_{1-\alpha }$ even when $\alpha <1/n$ where $n$ designates the total number of trials. In material sciences, this amounts to consider trials based on artificially modified materials; in the case when we aim at estimation of extreme upper quantiles, this amounts to strengthen the material. We would consider a family of increasing thresholds $%
x_{1},..,x_{m}$ and for each of them realize $K_{1},..,K_{m}$ trials, each block of iid realizations $Y$’s being therefore functions of the corresponding unobserved $R$’s with distribution $G$ conditioned upon $%
\{R>x_{l}\}$, $1\leq l\leq m.$ design which allows for the estimation of extreme quantiles.

The present setting is therefore quite different from that usually considered for similar problems under complete information. As sketched above it is specifically suited for industrial statistics 