dichotomous data
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In this section we review two standard methods in the industry and in biostatistics, which are the closest to our purpose. Up to our knowledge, no technique specifically addresses inference for extreme quantiles.

We address the estimation of small quantiles, hence the events of interest are of the form $\left( R<s\right) $ and the quantile is $q_{\alpha }$ for small $\alpha .$

The first method is the *staircase*, which is the present tool used to characterize a material fatigue strength*.*

The second one is the *Continual Reassessment Method (CRM)* which is adapted for assessing the admissible toxicity level of a drug in Phase 1 clinical trials.

Both methods rely on a parametric model for the distribution of the strength variable $R.$ We have considered two specifications, which allow for simple comparisons of performance, and do not aim at an accurate modelling in safety.

### The Staircase method

Denote $\mathbb{P}(R\leq s)=\phi (s,\theta _{0})$. Invented by Dixon and Mood (1948 [@dixon]), this technique aims at the estimation of the parameter $\theta _{0}$ through sequential search based on data of exceedances under thresholds. The procedure is as follows.

**Procedure**

Fix

-   The initial value for the constraint, $S_{ini}$,

-   The 