The randomization of a (t,m,s)-net or a (t,s)-sequence proposed by Owen (1995, Monte Carlo and quasi-Monte Carlo Methods in Scientific Computing (Berlin), Lecture Notes in Statistics, vol. 106, pp. 299-317) combines the strengths of equidistribution and Monte Carlo integration. This paper shows that the order of variance found in Owen (1997b, Scrambled net variance for integrals of smooth functions. Ann. Statist. 25, 1541-1562) still holds for weaker smooth integrands. The main results in this paper are as follows: For the univariate Lipschitz integrands on [0,1), the variance over scrambled ([lambda],t,m,1)-nets is of order O(n-3); In the s-dimensional case, if the integrand satisfies a generalized Lipschitz condition on [0,1)s, then the variance over scrambled ([lambda],t,m,s)-nets is of order O(n-3(log n)s-1); Also shown in this paper is that the variance over a scrambled (t,s)-sequence is of order O(n-2(log n)s-1) for the integrands described above. These results seem to be sharper than any similar results in the literature.
