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.
Highlights
Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods estimate the integral of a function f over the s-dimensional unit hypercube [0, 1)s = {u = (u1, . . . , us) : 0 ≤ uj < 1 for all j}, by evaluating f at n points in this hypercube and taking the average
We focus on randomly-shifted lattice rules, a widely-used and effective randomized quasi-Monte Carlo (RQMC) technique (L’Ecuyer and Lemieux, 2000; Sloan and Joe, 1994), where the n evaluation points are those of a randomly shifted integration lattice of density n that lie in the unit hypercube, in s dimensions
We explore and illustrate how the error behaves as a function of the random shift, and study its asymptotic distribution, in a variety of settings
Summary
For certain RQMC methods that involve a sufficient amount of randomization of the dependent points, a CLT has been proved to hold for n → ∞ Two such special cases are Latin hypercube sampling (LHS) (Owen, 1992), for which a bound on the total variation convergence to the normal distribution is available, and digital nets with the full nested scrambling of Owen (Loh, 2003). We show that for a sum of s non c-periodic one-dimensional functions, the cdf of n times the integration error converges to a spline (a piecewise-polynomial function) of degree s when n → ∞, and this limiting distribution has a bounded support (in contrast with the normal distribution) In this sense, the behavior is significantly different than when a classical CLT holds. Preliminary results (mostly for the one-dimensional case) were published in the proceedings of the 2009 Winter Simulation Conference (L’Ecuyer and Tuffin, 2009)
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