at the core of the motivation for this paper, and in order to make this approach more intuitive, we first sketch briefly the industrial situation which motivated this work in Section \[IndContext\]. We look at a safety property, namely thresholds $x$ which specify very rare events, typically failures under very small solicitation.

As stated above, the problem at hand is the estimation of very small quantiles. Classical techniques in risk theory pertain to large quantiles estimation. For example, the Generalized Pareto Distribution, to be referred to later on, is a basic tool in modeling extreme risks and exceedances over thresholds. Denoting $R$ the variable of interest and $\widetilde{R}:=1/R,$ then obviously, for $x>0$, $\left\{ R<x\right\} $ is equivalent to $\left\{
\widetilde{R}>u\right\} $ with $u=1/x$. In this paper we will therefore make use of this simple duality, stating formulas for $R,$ starting with classical results pertaining to $\widetilde{R}$ when necessary. Note that when $q_{\alpha }$ designates the $\alpha -$quantile of $R$ and respectively $\widetilde{q}_{1-\alpha }$ the $\left( 1-\alpha \right)-$quantile of $%
\widetilde{R}$, it holds $q_{\alpha }=1/\widetilde{q}_{1-\alpha }.$The resulting notation may seem a bit cumbersome; however the reader accustomed to industrial statistics will find it more familiar.

This article is organized as follows. 