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Nicolas Bousquet
Accelerated Monte Carlo estimation of exceedance probabilities under monotonicity constraints
Annales de la faculté des sciences de Toulouse Sér. 6, 21 no. 3 (2012), p. 557-591, doi: 10.5802/afst.1345
Article PDF | Analyses MR 3076412 | Zbl 1275.62058

Résumé - Abstract

On considère l’estimation de la probabilité ${p}=P(g({\bf X})\le 0)$ où $\bf X$ est un vecteur aléatoire et $g$ une fonction monotone. Premièrement, on rappelle et formalise une méthode, proposée par de Rocquigny (2009), permettant d’encadrer ${p}$ par des bornes déterministes en fonction d’un plan d’expérience séquentiel. Le second et principal apport de l’article est la définition et l’étude d’un estimateur statistique de ${p}$ tirant parti des bornes. Construit à partir de tirages uniformes successifs, cet estimateur présente sous de faibles conditions théoriques une variance asymptotique plus faible et une meilleure robustesse que l’estimateur classique de Monte Carlo, ce qui rend la méthode adaptée à l’emploi de codes informatiques $g$ lourds en temps de calcul. Des expérimentations numériques sont menées sur des exemples-jouets et un cas d’étude hydraulique plus réaliste. Une heuristique de boostrap, reposant sur un réplicat de l’hypersurface $\lbrace {\bf x}, \ g({\bf x})= 0\rbrace $ par des réseaux de neurones, est proposée et testée avec succès pour ôter le biais non-asymptotique de l’estimateur.

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