Risk measures in reliability engineering: comparison, duality and decision-making

In order to guarantee the reliability of industrial systems, statistical studies can be carried out on a variable of interest, Y, or, more precisely, on a risk measure computed based on this variable. This article studies and compares different risk measures, on the one hand, according to their theoretical properties, and, on the other hand, according to their numerical estimation. In particular, a distinction is proposed between risk measures homogeneous to the variable of interest Y (margin, quantile, and superquantile) and risk measures homogeneous to a probability (failure and buffered failure probability). More specifically, a duality between these two categories is developed, showing the equivalence of using a Y-homogeneous risk measure with a dual, homogeneous, [0,1]-homogeneous risk measure. Finally, a non-asymptotic statistical framework is proposed for decision-making using these risk measures. The latter can be used in practice in the case where the considered risk measures are the probability of failure and the quantile, and is equivalent, under one assumption, to a quantile estimation using the Wilks' method.