Variance of quadrature over scrambled unions of nets

Based on the work of Owen (1997a,b) who studied the variance of quadrature under a scrambled net with sample size n = lambda b(m), this paper investig ates scrambled sequences with sample sizes other than lambda b(m). First, t he variance of quadrature under a scrambled sequence which is a union of tw o nets in base b is found. The scrambling schemes applied to the two nets c an be independent or simultaneous. The results can be extended to the union of more than two nets. For finite sample sizes, the scrambled net-union va riance is bounded by a small constant multiple of the Monte Carlo variance. Second, it is shown that for any Lipschitz integrand on [0,1), the varianc e is O(n(-3)) for a scrambled net, and O(n(-3+alpha)) for a union of two sc rambled nets in base b, for a certain alpha is an element of [0, 1]. For an y multivariate smooth integrand on [0, 1)(s), the scrambled net-union varia nce is O(n(-3+alpha)(log n)((s-1)1 alpha<1)) for a certain alpha is an elem ent of [0, 1]. It turns out that adding some additional points may sometime s cause a large loss of efficiency.