La Trobe University
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 talk, 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 \(n\to\infty\), 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.
Pierre L'Ecuyer is a Professor in the Departement d'Informatique et de Recherche Operationnelle at the Universite de Montreal. He currently holds the Canada Research Chair in Stochastic Simulation and Optimization and an Inria International Chair (at Inria-Rennes) for 2013-2018. He obtained the Steacie Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC) in 1995-97, twice the INFORMS Simulation Society Outstanding Research Publication Award, in 1999 and 2009, the Distinguished Service Award in 2011, a Killam Research Fellowship in 2001-03, the Urgel-Archambault Prize from ACFAS in 2002, and was elected INFORMS Fellow in 2006. He has published over 230 scientific articles and book chapters in various areas, including random number generation, quasi-Monte Carlo methods, efficiency improvement in simulation, sensitivity analysis and optimization for discrete-event simulation models, simulation software, stochastic dynamic programming, and applications in finance, manufacturing, telecommunications, reliability, and service center management. He also developed software libraries and systems for the theoretical and empirical analysis of random number generators and quasi-Monte Carlo point sets, and for general discrete-event simulation. His work impinges on the areas of mathematics, statistics, operations research, economics, and computer science. He is currently Associate Editor for ACM Transactions on Mathematical Software, Statistics and Computing, Cryptography and Communications, and International Transactions in Operational Research. He was Editor-in-Chief for the ACM Transactions on Modeling and Computer Simulation until June 2013. He has been a referee for over 120 different scientific journals. He was a professor in the Departement d'Informatique at Universite Laval (Quebec) from 1983 to 1990 and is at the Universite de Montreal since then. He has been a visiting scholar (for several months) at Stanford University (USA), INRIA-Rocquencourt (France), Ecole des Mines (France), Waseda University (Tokyo), University of Salzburg (Austria), North Carolina State University (USA), and INRIA-Rennes (France). He is a member of the CIRRELT and GERAD research centers, in Montreal.
Contact Maaike Wienk at email@example.com, with a cc to Darren Condon at La Trobe (D.Condon@latrobe.edu.au), one week in advance at the latest.
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