is recovered rather accurately, whereas the estimation of the standard deviation suffers a loss in accuracy, which in turn yields a relative error of 180 %  on the $10^{-3}$ quantile.

**Drawback of the Staircase method**

A major advantage of the Staircase lies in the fact that the number of trials to be performed in order to get a reasonable estimator of the mean is small. However, as shown by the simulations, this method is not adequate for the estimation of extreme quantiles. Indeed, the latter follows from an extrapolation based on estimated parameters, which furthermore may suffer of bias. Also, reparametrization of the distribution making use of the theoretical extreme quantile would not help, since the estimator would inherit of a large lack of accuracy.

### The Continuous Reassesment Method (CRM)

**General principle**

The CRM (O’Quigley, Pepe and Fisher, 1990[@quigley]) has been designed for clinical trials and aims at the estimation of $q_{\alpha }$ among $J$ stress levels $s_{1},...,s_{J}$, when $\alpha$ is of order $20\%$.

Denote $\mathbb{P}(R\leq s)=\psi (s,\beta _{0})$. The estimator of $q_{\alpha }$ is $$s^{\ast }:=\underset{s_j \in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
(s_{j},\beta _{0})-\alpha |}.$$This optimization is performed iteratively and $K$ trials are performed at each iteration.

Start with an initial estimator 