Estimation of extreme quantiles in a simulation model

ABSTRACT

A simulation model with an outcome $Y=m\left(X\right)$ is considered, where X is an ${\mathbb{R}}^d$-valued random variable and $m:{\mathbb{R}}^d\to \mathbb{R}$ is a smooth function. Estimates of the ${\alpha }_n$-quantile $q_{m\left(X\right),{\alpha }_n}$ of $m\left(X\right)$ 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 ${\alpha }_n\to 1$ $(n\to \mathrm{\infty })$ and $n\cdot (1-{\alpha }_n)\to 0$ $(n\to \mathrm{\infty })$. Finite sample behaviour of the estimates is illustrated by simulations.

KEYWORDS:

• Extreme quantiles
• nonparametric quantile estimation
• importance sampling
• rates of convergence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 62G05
• secondary 62G30

Acknowledgments

The authors would like to thank two referees and an associate editor for various comments which helped to improve the first version of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors would like to thank the German Research Foundation (DFG) for funding this project within Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 57157498 – SFB 805 and the Natural Sciences and Engineering Research Council of Canada for additional support under Grant RGPIN-2015-06412.