Statistica Sinica 9(1999), 451-473

VARIANCE OF QUADRATURE OVER SCRAMBLED

UNIONS OF NETS

Rong-Xian Yue

Hong Kong Baptist University and East China Normal University

Abstract: Based on the work of Owen (1997a,b) who studied the variance of quadrature under a scrambled net with sample size n=λb^m, this paper investigates scrambled sequences with sample sizes other than λb^m. First, the variance of quadrature under a scrambled sequence which is a union of two nets in base b is found. The scrambling schemes applied to the two nets can 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 variance 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 variance is O(n^-3) for a scrambled net, and O(n^-3+α) for a union of two scrambled nets in base b, for a certain . For any multivariate smooth integrand on [0,1)^s, the scrambled net-union variance is for a certain . It turns out that adding some additional points may sometimes cause a large loss of efficiency.

Key words and phrases: Integration, multiresolution, quasi-Monte Carlo.