Mathematics – Statistics Theory
Scientific paper
2010-12-05
Mathematics
Statistics Theory
23 pages. Extended version of a preprint under submission
Scientific paper
This article deals with the estimation of a probability p of an undesirable event. Its occurence is formalized by the exceedance of a threshold reliability value by the unidimensional output of a (time-consuming) computer code G with multivariate probabilistic input X. When G is assumed monotonous with respect to X, the Monotonous Reliability Method was proposed by de Rocquigny (2009) in an engineering context to provide bounds and crude estimates of p, via deterministic or stochastic designs of experiments. The present article consists in a formalization and technical deepening of this idea, as a large basis for future theoretical and applied studies. Three kinds of results are especially emphasized. First, the bounds themselves remain too crude and conservative estimators of p for a dimension of X upper than 2. Second, a maximum-likelihood estimator of p can be easily built, presenting a high variance reduction with respect to a standard Monte Carlo case, but suffering from conservative bias. Third, the theoretical properties of a family of unbiased estimators of p, based on sequential nested importance samplings, are analyzed. Their supplementary potential improvement requires further studies whose main lines are discussed. Along the paper, the efficiency and difficulties of these approaches are illustrated by a generic example. In fine, we show that both approaches lead to promising parsimonious estimation algorithms provided a sequential emulation of the limit state (failure) surface, seen as a supervised classification problem, can be made under monotony constraints. Besides, some connections and research avenues are identified in various mathematical areas like multivariate statistics, multi-objective optimization and computational geometry.
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