The mean square quasi-monte carlo error for digitally shifted digital nets

Abstract

In this paper, we study randomized quasi-Monte Carlo (QMC) integration using digitally shifted digital nets.We express the mean square QMC error of the nth discrete approximation fn of a function f: [0, 1)s → R for digitally shifted digital nets in terms of the Walsh coefficients of f. We then apply a bound on theWalsh coefficients for sufficiently smooth integrands to obtain a quality measure called Walsh figure of merit for the root mean square error, which satisfies a Koksma-Hlawka type inequality on the root mean square error. Through two types of experiments, we confirm that our quality measure is of use for finding digital nets which show good convergence behavior of the root mean square error for smooth integrands.