On Dropping the First Sobol’ Point

Abstract

Quasi-Monte Carlo (QMC) points are a substitute for plain Monte Carlo (MC) points that greatly improve integration accuracy under mild assumptions on the problem. Because QMC can give errors that are o(1/n) as $$n\rightarrow \infty $$ , and randomized versions can attain root mean squared errors that are o(1/n), changing even one point can change the estimate by an amount much larger than the error would have been and worsen the convergence rate. As a result, certain practices that fit quite naturally and intuitively with MC points can be very detrimental to QMC performance. These include thinning, burn-in, and taking sample sizes such as powers of 10, when the QMC points were designed for different sample sizes. This article looks at the effects of a common practice in which one skips the first point of a Sobol’ sequence. The retained points ordinarily fail to be a digital net and when scrambling is applied, skipping over the first point can increase the numerical error by a factor proportional to $$\sqrt{n}$$ where n is the number of function evaluations used.