On the distribution of integration error by randomly-shifted lattice rules

Abstract

A lattice rule with a randomly-shifted lattice estimates a mathematical expectation, written as an integral over the s-dimensional unit hypercube, by the average of n evaluations of the integrand, at the n points of the shifted lattice that lie inside the unit hypercube. This average provides an unbiased estimator of the integral and, under appropriate smoothness conditions on the integrand, it has been shown to converge faster as a function of n than the average at n independent random points (the standard Monte Carlo estimator). In this paper, we study the behavior of the estimation error as a function of the random shift, as well as its distribution for a random shift, under various settings. While it is well known that the Monte Carlo estimator obeys a central limit theorem when n→∞, the randomized lattice rule does not, due to the strong dependence between the function evaluations. We show that for the simple case of one-dimensional integrands, the limiting error distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. We find that in higher dimensions, there is little hope to precisely characterize the limiting distribution in a useful way for computing confidence intervals in the general case. We nevertheless examine how this error behaves as a function of the random shift from different perspectives and on various examples. We also point out a situation where a classical central-limit theorem holds when the dimension goes to infinity, we provide guidelines on when the error distribution should not be too far from normal, and we examine how far from normal is the error distribution in examples inspired from real-life applications.