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June 2011 Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands
Josef Dick
Ann. Statist. 39(3): 1372-1398 (June 2011). DOI: 10.1214/11-AOS880

Abstract

We study a random sampling technique to approximate integrals [0, 1]sf(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics.

The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the number of samples) for functions f with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order N−3/2+ε under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order N−3/2−1/s+ε (where s ≥ 1 is the dimension) for functions with mixed partial derivatives up to order two.

Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate.

This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order Nα−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 1 in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order N−5/2 and N−7/2, in accordance with the theoretical upper bound.

Citation

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Josef Dick. "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands." Ann. Statist. 39 (3) 1372 - 1398, June 2011. https://doi.org/10.1214/11-AOS880

Information

Published: June 2011
First available in Project Euclid: 4 May 2011

zbMATH: 1298.65011
MathSciNet: MR2850206
Digital Object Identifier: 10.1214/11-AOS880

Subjects:
Primary: 65C05
Secondary: 65D32

Keywords: Digital nets , quasi-Monte Carlo , randomized quasi-Monte Carlo

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 3 • June 2011
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