MONTE-CARLO VARIANCE OF SCRAMBLED NET QUADRATURE

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. This paper studies the varian ce of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. The integrand is assumed to be measurable and square integrable but not necessarily of bounded variation. In simple Monte Carlo, every nonconstant term of th e multiresolution contributes to the variance of the estimated integra l. For scrambled nets, certain low-dimensional and coarse terms do not contribute to the variance. For any integrand in L-2, the sampling va riance tends to zero faster under scrambled net quadrature than under Monte Carlo sampling, as the number of function evaluations n tends to infinity. Some finite n results bound the variance under scrambled ne t quadrature by a small constant multiple of the Monte Carlo variance, uniformly over all integrands f. Latin hypercube sampling is a specia l case of scrambled net quadrature.