Estimation of extreme quantiles in a simulation model

Michael Kohler and Adam Krzyżak

Journal of Nonparametric Statistics, 2019, vol. 31, issue 2, 393-419

Abstract: A simulation model with an outcome $ Y=m(X) $ Y=m(X) is considered, where X is an $ {\mathbb {R}^d} $ Rd-valued random variable and $ m: {\mathbb {R}^d} \rightarrow \mathbb{R} $ m:Rd→R is a smooth function. Estimates of the $ \alpha _n $ αn-quantile $ q_{m(X),\alpha _n} $ qm(X),αn of $ m(X) $ m(X) based on surrogate model of m and on importance sampling are constructed which use at most n evaluations of the function m. Results concerning the rate of convergence of the estimates are derived in case that $ \alpha _n \rightarrow 1 $ αn→1 $ (n \rightarrow \infty ) $ (n→∞) and $ n \cdot (1-\alpha _n) \rightarrow 0 $ n⋅(1−αn)→0 $ (n \rightarrow \infty ) $ (n→∞). Finite sample behaviour of the estimates is illustrated by simulations.

Date: 2019
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DOI: 10.1080/10485252.2019.1567727

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