on the endurance limit, which is also the hardest regime to characterize since there is usually only few and scattered observations.

In this framework, we are focusing on minimal risk. The critical quantities that are used to characterize minimal risk linked to fatigue damage are failure quantiles, called in this framework allowable stresses at a given number of cycles and for a fixed level of probability. Those quantiles are of great importance since they intervene in decisions pertaining engine parts dimensioning, pricing decisions as well as maintenance policies.

Formalization of the industrial problem
---------------------------------------

The aim of this study is to propose a new design method for the characterization of allowable stress in very high cycle fatigue, for a very low risk $\alpha $ of order $10^{-3}$. We are willing to obtain a precise estimation method of the $\alpha -$failure quantile based on a minimal number of trials.

Denote $N$ the lifetime of a material in terms of number of cycles to failure and $S$ the stress amplitude of the loading, in MPa. Let $n_{0}$ be the targeted time span of order $10^{6}-10^{7}$ cycles.

Define the allowable stress $s_{\alpha }$ at $n_{0}$ cycles and level of probability $\alpha $ $=10^{-3}$ 