Achieving High Convergence Rates by Quasi-Monte Carlo and Importance Sampling for Unbounded Integrands

We consider the problem of estimating an expectation E[h(X)] by quasi-Monte Carlo (QMC) methods, where h:R-d -> R is an unbounded smooth function and X is a standard normal random vector. While the classical Koksma-Hlawka inequality cannot be directly applied to unbounded functions, we establish a novel framework to study the convergence rates of QMC for unbounded smooth integrands. We propose a projection method to modify the unbounded integrands into bounded and smooth ones, which differs from the low variation extension strategy of avoiding the singularities along the boundary of the unit cube [0,1](d) in Owen [SIAM Rev., 48 (2006), pp. 487-503]. The total error is then bounded by the quadrature error of the transformed integrand and the projection error. We prove that if the function h(x) and its mixed partial derivatives do not grow too fast as the Euclidean norm |x| tends to infinity, then projection-based QMC and randomized QMC (RQMC) methods achieve an error rate of O(n(-1+epsilon)) with a sample size n and an arbitrarily small epsilon>0 . However, the error rate turns out to be only O(n(-1+2M+epsilon)) when the functions grow exponentially as O(exp{M|x|(2)}) with M is an element of(0,1/2) . Remarkably, we find that using importance sampling with t -distribution as the proposal can dramatically improve the root mean squared error of RQMC from O(n(-1+2M+epsilon)) to O(n(-3/2+epsilon)) .