interest and the latter aims at the estimation of quantiles of order 0.2 or so, far from the target $\alpha =10^{-3}$. Therefore, we propose an original procedure designed for the estimation of extreme quantiles under binary information.

![Relative error on the $10^{-3}$-quantile with respect to the number of trials for each stress level[]{data-label="crm"}](erreursCRM_m_tronc2.png)

A new design for the estimation of extreme quantiles {#Splitting}
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Splitting
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The design we propose is directly inspired by the general principle of Splitting methods used in the domain of rare events simulation and introduced by Kahn and Harris (1951 [@Kahn1951]).

The idea is to overcome the difficulty of targeting an extreme event by decomposing the initial problem into a sequence of less complex estimation problem. This is enabled by the splitting methodology which decompose a small probability into the product of higher order probabilities.

Denote $\mathbb{P}$ the distribution of the r.v. $R$. The event $\{ R\le s_\alpha \}$ can be expressed as the intersection of inclusive events for $s_{\alpha }=s_{m}<s_{m-1}<...<s_{1}$ it holds: $$\{R\leq s_{\alpha }\}=\{R\leq s_{m}\}\subset \dots \subset \{R\leq s_{1}\}.$$

It follows that

$$\mathbb{P}(R\leq s_{\alpha })=\mathbb{P}(R\leq
s_{1})\prod_{j=1}^{m-1}\mathbb{P}(R\leq s_{j+1}\mid R\leq s_{j})
\label{Prod}$$

\[split\]

The thresholds $(s_{j})_{j=1,\dots ,m}$ should be chosen such that all $\mathbb{P}(R\leq s_{j+1}\mid 