$\widehat{\beta _{1}}$ of $\beta _{0}$, for example through a Bayesian choice as proposed in [@quigley]. Define $$s_{1}^{\ast }:=\underset{s_j \in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
(s_{j},\widehat{\beta _{1}})-\alpha |}.$$

Every iteration follows a two-step procedure:

**Step 1.** Perform $J$ trials under $\psi (.,\beta _{0})$, say $R_{1,1},..,R_{1,J}$ and observe only their value under threshold, say $Y_{1,j}:={\Large 1}_{R_{1,j}<s_{1}^{\ast }},1\leq j\leq J.$

**Step i.** Iteration $i$ consists in two steps :

-   Firstly an estimate $\widehat{\beta _{i}}$ of $\beta _{0}$ is produced on the basis of the information beared by the trials performed in all the preceding iterations through maximum likelihood under $\psi (.,\beta _{0})$ (or by maximizing the posterior distribution of the parameter).

-   $$s_{i}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|\psi
    (s,}\widehat{{\beta _{i}}}{)-\alpha |};$$

    This stress level $s_{i}^{\ast }$ is the one under which the next $K$ trials $Y_{i,1},\dots,Y_{i,K}$ will be performed in the Bernoulli scheme $\mathcal{B}\left(\psi (s_{i}^{\ast },\beta _{0})\right)$.

The stopping rule depends on the context (maximum number of trials or stabilization of the results).

Note that the bayesian inference is useful in the cases where there is no diversity in the observations at some iterations of the procedure, i.e when, at a given level of test $s_i^*$, only failures or survivals are observed.

**Application to fatigue data**

The application to 