On Dropping the First Sobol’ Point

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