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 alpha, an optimal QMC rule converges at a best-possible rate O(N-alpha/d). However, in applications the value of alpha can be unknown and/or a rate-optimal QMC rule can be unavailable. Standard practice is to employ alpha(L)-optimal QMC where the lower bound alpha(L) <= alpha 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.