Scrambling Sobol' and Niederreiter–Xing Points

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. In particular, randomized (0, m , s )-nets in base b produce unbiased estimates of the integral, have a variance that tends to zero faster than 1/ n for any square integrable integrand and have a variance that for finite n is never more than e ≐2.718 times as large as the Monte Carlo variance. Lower bounds than e are known for special cases. Some very important ( t , m , s )-nets have t >0. The widely used Sobol' sequences are of this form, as are some recent and very promising nets due to Niederreiter and Xing. Much less is known about randomized versions of these nets, especially in s >1 dimensions. This paper shows that scrambled ( t , m , s )-nets enjoy the same properties as scrambled (0, m , s )-nets, except the sampling variance is guaranteed only to be below b t [( b +1)/( b −1)] s times the Monte Carlo variance for a least-favorable integrand and finite n .