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.