Scrambled net variance for integrals of smooth functions

Abstract

Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. One version, randomized $(t, m, s)$-nets, has the property that the integral estimates are unbiased and that the variance is $o(1/n)$, for any square integrable integrand.

Stronger assumptions on the integrand allow one to find rates of convergence. This paper shows that for smooth integrands over s dimensions, the variance is of order $n^{-3}(\log n)^{s-1}$, compared to $n^{-1}$ for ordinary Monte Carlo. Thus the integration errors are of order $n^{-3/2}(\log n)^{(s-1)/2} in probability. This compares favorably with the rate $n^{-1}(\log n)^{s-1}$ for unrandomized $(t, m, s)$-nets.