Extensible grid sampling for quantile estimation

Quantiles are used as a measure of risk in many stochastic systems. We study the estimation of quantiles with the Hilbert space-filling curve (HSFC) sampling scheme that transforms specifically chosen one-dimensional points into high dimensional stratified samples while still remaining the extensibility. We study the convergence and asymptotic normality for the estimate based on HSFC. By a generalized Dvoretzky–Kiefer–Wolfowitz inequality for independent but not identically distributed samples, we establish the strong consistency for such an estimator. We find that under certain conditions, the distribution of the quantile estimator based on HSFC is asymptotically normal. The asymptotic variance is of O ( n − 1 − 1 / d ) O(n^{-1-1/d}) when using n n HSFC-based quadrature points in dimension d d , which is more efficient than the Monte Carlo sampling and the Latin hypercube sampling. Since the asymptotic variance does not admit an explicit form, we establish an asymptotically valid confidence interval by the batching method. We also prove a Bahadur representation for the quantile estimator based on HSFC. Numerical experiments show that the quantile estimator is asymptotically normal with a comparable mean squared error rate of randomized quasi-Monte Carlo (RQMC) sampling. Moreover, the coverage of the confidence intervals constructed with HSFC is better than that with RQMC.