SCS

Abstract

Let $X$ be a random vector with distribution $\mu$ on ${\mathbb R}^d$ and $\Phi$ be a mapping from ${\mathbb R}^d$ to ${\mathbb R}$. That mapping acts as a black box, e.g., the result from some computer experiments for which no analytical expression is available. This paper presents an efficient algorithm to estimate a tail probability given a quantile or a quantile given a tail probability. It proceeds by successive elementary steps, each one being based on Metropolis-Hastings algorithm. The algorithm improves upon existing multilevel splitting methods and can be analyzed using Poisson process tools that lead to exact description of the distribution of the estimated probabilities and quantiles. The performance of the algorithm is demonstrated in a problem related to digital watermarking.