$q_{1-\alpha }$, the $\left( 1-\alpha \right) $-quantile of $G$ when $\alpha $ is much smaller than $1/n.$ We therefore aim at the estimation of so-called extreme quantiles. This question has been handled by various authors, and we will review their results somehow later. The approach which we develop is quite different since we do not assume that the $X_{i}$’s  can be observed. For any threshold $x$, we define the r.v. $$Y=\left\{ 
\begin{array}{l}
1\text{ if }X\leq x \\ 
0\text{ if }X>x%
\end{array}%
\right.$$which therefore has a Bernoulli distribution with parameter $G(x).$ We may choose $x$, however we do not observe $X$, but merely $Y.$ Therefore any inference on $G$ suffers from a severe loss of information. This kind of setting is common in industrial statistics: When exploring the strength of a material, or of a bundle, we may set a constraint $x$, and observe whether the bundle breaks or not when subjected at this level of constraint.

In the following, we will denote $R$ the resistance of this material, we observe $Y.$ Inference on $G$ can be performed for large $%
n$ making use of many thresholds $x.$ Unfortunately such a procedure will not be of any help for extreme quantiles. 