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We consider the problem of estimating the -quantile for a given functional evaluated on solutions of a deterministic model in which model input is subject to stochastic variation. We derive upper and lower bounding estimators of the -quantile. We perform an a posteriori error analysis for the -quantile estimators that takes into account the effects of both the stochastic sampling error and the deterministic numerical solution error and yields a computational error bound for the estimators. We also analyze the asymptotic convergence properties of the -quantile estimator bounds in the limit of large sample size and decreasing numerical error and describe algorithms for computing an estimator of the -quantile with a desired accuracy in a computationally efficient fashion. One algorithm exploits the fact that the accuracy of only a subset of sample values significantly affects the accuracy of a -quantile estimator resulting …