$s_{i}$ is the stress applied on specimen $i.$

Note that the number of observations is constrained by industrial and financial considerations; Thus $\alpha $ is way lower than $1/n$ and we are considering a quantile lying outside the sample range.

While we motivate this paper with the above industrial application, note that this kind of problem is of interest in other domains, such as broader reliability issues or medical trials through the estimation of the maximum tolerated dose of a given drug.

Extreme quantile estimation, a short survey {#revLit}
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As seen above estimating the minimal admissible constraint raises two issues; on one hand the estimation of an extreme quantile, and on the other hand the need to proceed to inference based on exceedances under thresholds.  We present a short exposition of these two areas, keeping in mind that the literature on extreme quantile estimation deals with complete data, or data under right censoring.

Extreme quantiles estimation methods
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Extreme quantile estimation in the univariate setting is widely covered in the literature when the variable of interest $X$ is either completely or partially observed.

The usual framework is the study of the $(1-\alpha)-$quantile of a r.v $X$, with very small 