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Control Functionals for Quasi-Monte Carlo Integration

Quasi-Monte Carlo (QMC) methods are being adopted in statistical applications due to the increasingly challenging nature of numerical integrals that are now routinely encountered. For integrands with $d$-dimensions and derivatives of order $α$, an optimal QMC rule converges at a best-possible rate $O(N^{-α/d})$. However, in applications the value of $α$ can be unknown and/or a rate-optimal QMC rule can be unavailable. Standard practice is to employ $α_L$-optimal QMC where the lower bound $α_L \leq α$ is known, but in general this does not exploit the full power of QMC. One solution is to trade-off numerical integration with functional approximation. This strategy is explored herein and shown to be well-suited to modern statistical computation. A challenging application to robotic arm data demonstrates a substantial variance reduction in predictions for mechanical torques.