How to attach a trustworthy error bar to a quasi-Monte Carlo (QMC) integral estimate. Plain Monte Carlo reports error through the sample variance and the central limit theorem, but deterministic QMC point sets have no built-in statistical error estimate. This topic covers the two families of remedies: deterministic error certificates (bounds from bounded variation or complete monotonicity of the integrand) and randomized QMC, where scrambling and random digital shifts create independent unbiased replicates whose spread gives a practical confidence interval. It also covers modern estimators such as median-of-means and bootstrapping the replicates, and the subtlety that some randomized QMC estimates are nearly symmetric enough to justify intervals without invoking a central limit theorem.
A tutorial survey of how to quantify the error in quasi-Monte Carlo (QMC) integral estimates. Where plain Monte Carlo gets its error bar from the sample variance and the central limit theorem, determ…