Randomized quasi-Monte Carlo integration

Abstract

Quasi-Monte Carlo sampling is a numerical integration method that uses points with a space-filling property in $[0,1{]}^s$ designed to give better estimates than plain Monte Carlo methods do. For integrands of bounded variation in the sense of Hardy and Krause, errors of $O(n^{-1+\epsilon })$ for any $\epsilon >0$ are obtained from $n$ sample points. Randomized quasi-Monte Carlo (RQMC) points are individually uniformly distributed but collectively space-filling and then independent replications provide variance estimates. For smooth enough integrands the randomization can give a root mean squared error of $O(n^{-3/2+\epsilon })$. This article explains RQMC for a statistical readership recounting some history and presenting some current directions.