the posterior distribution of the parameter:

$$\pi^*_{s_i} \sim \beta \left(k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}~,~ n + (J\times i) -(k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}) +1 \right)$$

The above distribution also corresponds an order statistic of the uniform distribution $U_{k + \sum_{l=1}^{i}\sum_{j=1}^{J}Y_{l,j}~,~n + (J\times i) }$. We then obtain an estimate $\Lambda_{s_1^\ast}$.

The next stress level $s_{i+1}^{\ast }$ to be tested in the procedure is then given by $$s_{i+1}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|( 1 - \exp(- \Lambda_{s_1^\ast} s_j))-\alpha |}.$$

**Numerical simulation for the CRM**

Under the exponential model with parameter $\lambda =0.2$ and through $N=10$ iterations of the procedure, and $J=10$, with equally distributed thresholds $s_{1},..,s_{J}$ , and performing $K=50$ trials at each iteration, the results in Table \[CRMexp\] are obtained.

[|c|c|c|c|]{}\
&\
Mean & Std & Mean & Std\
0.129 & 0.48 & -0.799 & 0.606\

The $10^{-3}-$quantile is poorly estimated on a fairly simple model. Indeed for thresholds close to the expected quantile, nearly no failure is observed. So, for acceptable $K$, the method is not valid; figure \[crm\] shows the increase of accuracy with respect to $K.$

Both the Staircase and the CRM have the same drawback in the context of extreme quantile estimation, since the former targets the central tendency of the variable of 