QUASI-RANDOM METHODS FOR ESTIMATING INTEGRALS USING RELATIVELY SMALL SAMPLES

Much of the recent work dealing with quasi-random methods has been aim ed at establishing the best possible asymptotic rates of convergence t o zero of the error resulting when a finite-dimensional integral is re placed by a finite sum of intergrand values. In contrast with this per spective to concentrate on asymptotic convergence rates, this paper em phasizes quasi-random methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finite-dimensional in tegrals is used to illustrate the major ideas, although much of what i s done applies equally to the problem of solving certain Fredholm inte gral equations. Some new techniques, based on error-reducing transform ations of the integrand, are described that have been shown to be usef ul both in estimating high-dimensional integrals and in solving integr al equations. These techniques illustrate the utility of carrying over to the quasi-Monte Carlo method certain devices that have proven to b e very valuable in statistical (pseudorandom) Monte Carlo applications .