We study numerical approximations of integrals by averaging the function at some sampling points. Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order (where is the number of samples). Quasi-Monte Carlo (QMC) sampling on the other hand achieves a convergence of order , for any . Randomized QMC (RQMC), a combination of MC and QMC, achieves a RMSE of order . A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order (where is the dimension). QMC, RQMC and RQMC with local antithetic sampling require that the integrand has some smoothness (for instance, bounded variation). Stronger smoothness assumptions on the integrand do not improve the convergence of the above algorithms further. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the RMSE of order if the integrand has square integrable partial mixed derivatives up to order in each variable. Known lower bounds show that this rate of convergence cannot be improved. We provide numerical examples for which the RMSE converges approximately with order and , in accordance with the theoretical upper bound.
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