Quasi-Random Methods for Estimating Integrals Using Relatively Small Samples

Much of the recent work dealing with quasi-random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite-dimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasi-random methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finite-dimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on error-reducing transformations of the integrand, are described that have been shown to be useful both in estimating high-dimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasi-Monte Carlo method certain devices that have proven to be very valuable in statistical (pseudorandom) Monte Carlo applications.