Adaptive Estimation of Quantiles in a Simulation Model

Let X be an R
^d
valued random variable, let m : R
^d
→ R be a measurable function and set Y = m(X). Given a sample of (X, Y) of size n, we consider the problem of estimating the quantile of Y of a given level α ∈ (0, 1). A method for choosing the parameter of a surrogate model of m is introduced, and it is shown that the corresponding surrogate quantile estimate achieves the rate of convergence bounded by the sum of the minimal rate of convergence of the quantile estimates corresponding to the given surrogate estimates and a term of order log(n)/n. The finite sample size behavior of this quantile estimate is illustrated by applying it to simulated data and to a quantile estimation problem in mechanical engineering.