Nonparametric Quantile Estimation Based on Surrogate Models

Nonparametric estimation of a quantile q m(X),α of a random variable m(X) is considered, where m : ℝ d → ℝ is a function, which is costly to compute and X is an ℝ d -valued random variable with known distribution. Monte Carlo surrogate quantile estimates are considered, where in a first step, the function m is estimated by some estimate (surrogate) m n , and then, the quantile q m(X),α is estimated by a Monte Carlo estimate of the quantile qm n(X),α . A general error bound on the error of this quantile estimate is derived, which depends on the local error of the function estimate m n , and the rates of convergence of the corresponding Monte Carlo surrogate quantile estimates are analyzed for two different function estimates. The finite sample size behavior of the estimates is investigated in simulations.