the level of stress that guarantee that the risk of failure before $n_{0}$ does not exceed $\alpha $: $$s_{\alpha }=\sup \left\{ s:\mathbb{P}(N\leq n_{0}|S=s)\leq \alpha \right\} 
\label{pb1}$$

We will now introduce a positive r.v. $R=R_{n_{0}}$ modeling the resistance of the material at $n_{0}$ cycles and homogeneous to the stress. $R$ is the variable of interest in this study and its distribution $\mathbb{P}$ is defined as: $$\mathbb{P}(R\leq s) = \mathbb{P}(N\leq n_{0}|S=s).  \label{loiR}$$

Thus, the allowable stress can be rewritten as the $\alpha-$quantile of the distribution of $R$,

$$s_{\alpha }=q_{\alpha }=\sup \left\{ s:\mathbb{P}(R\leq s)\leq \alpha \right\}.$$

However, $R$ is not directly observed. Indeed, the usable data collected at the end of a test campaign consists in couples of censored fatigue life - stress levels $\left(\min (N,n_{0}),s\right)$ where $s$ is part of the design of the experiment. The relevant information that can be drawn from those observations to characterize $R$ is restricted to indicators of whether or not the specimen tested has failed at $s$ before $n_{0}$. Therefore, the relevant observations corresponding to a campaign of $n$ trials are formed by a sample of variables $Y_{1},...,Y_{n}$ with for $1\leq i\leq n,$ $$Y_{i}=\left\{ 
\begin{array}{l}
1\text{ if }R_{i}\leq s_{i} \\ 
0\text{ if }R_{i}>s_{i}%
\end{array}%
\right.$$

where 