Estimating a probability of failure with the convex order in computer experiments

This paper deals with methods for estimating the probability of failure of an industrial product. To be more specific, it is defined as the probability that the output of a physical model, with random input variables, exceeds a threshold. The model corresponds with an expensive to evaluate black-box function, so that the classical Monte Carlo simulation methods cannot be put into practice. Basic Bayesian principles of the so-called Kriging method are then applied to correctly design an estimator of the target probability. From a numerical point of view, the practical use of this estimator is unfortunately restricted. An alternative estimator is then considered, which is equivalent in term of bias. The main result of this paper concerns the existence of a convex order inequality between these two estimators. We show that this inequality allows to compare their efficiency and to quantify the uncertainty on the estimation results they provide. A sequential procedure for the construction of an optimal design of computer experiments, based on the principle of the Stepwise Uncertainty Reduction strategies, also results from this inequality. The interest of this approach is highlighted through the study of a real case from the company STMicroelectronics