step $\delta >0$,

-   The number of cycles $n_0$ to perform before concluding a trial,

-   The total number of items to be tested, $K$.

The first item is tested at level $s_{(1)}=S_{ini}$. The next item is tested at level $s_{(2)}=S_{ini}-\delta$ in case of failure and $s_{(2)}=S_{ini}+\delta$ otherwise. Proceed sequentially on the $K-2$ remaining specimen at a level increased (respectively decreased) by $\delta$ in case of survival (resp. failure). The process is illustrated in figure \[staircase\].

Note that the proper conduct of the Staircase method relies on strong assumptions on the choice of the design parameters. Firstly, $S_{ini}$ has to be sufficiently close to the expectation of $R$ and secondly, $\delta $ has to lay between $0.5\sigma $ and $2\sigma $, where $\sigma$ designates the standard deviation of the distribution of $R$.

Denote $\mathbb{P}(R\leq s)=\phi (s,\theta _{0})$ and $Y_{i}$ the variable associated to the issue of the trial $i$, $1\leq
i\leq K$, where $Y_{i}$ takes value $1$ under failure and $0$ under no failure, $Y_{i}=\mathds{1}_{N_{a}\leq n_{0}}\sim \mathcal{B}(\phi
(s_{i},\theta _{0}))$.

![image](staircaseim.png) \[staircase\]

**Estimation**

After the $K$ trials, the parameter $\theta _{0}$ is estimated through maximization of the likelihood, namely

$$\widehat \theta = \underset{\theta}{\text{argmax}}{\prod_{i=1}^K
\phi(s_i,\theta)^{y_i } (1-\phi(s_i,\theta))^{ (1-y_i) }}.$$

**Numerical results**

The 