Electronic Journal of Statistics

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

Pierre L’Ecuyer, David Munger, and Bruno Tuffin

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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.

Article information

Electron. J. Statist., Volume 4 (2010), 950-993.

First available in Project Euclid: 6 October 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Quasi-Monte Carlo lattice rule integration error limit theorem confidence interval


L’Ecuyer, Pierre; Munger, David; Tuffin, Bruno. On the distribution of integration error by randomly-shifted lattice rules. Electron. J. Statist. 4 (2010), 950--993. doi:10.1214/10-EJS574. https://projecteuclid.org/euclid.ejs/1286371471

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  • Abramowitz, M. and Stegun, I. A. (1970)., Handbook of Mathematical Functions. Dover, New York.
  • Afflerbach, L. and Grothe, H. (1985). Calculation of Minkowski-Reduced Lattice Bases., Computing 35 269–276.
  • Avramidis, A. N. and Wilson, J. R. (1996). Integrated Variance Reduction Strategies for Simulation., Operations Research 44 327-346.
  • Avramidis, A. N. and Wilson, J. R. (1998). Correlation-Induction Techniques for Estimating Quantiles in Simulation Experiments., Operations Research 46 574–591.
  • Barrow, D. L. and Smith, P. W. (1979). Spline Notation Applied to a Volume Problem., The American Mathematical Monthly 86 50–51.
  • Caflisch, R. E., Morokoff, W. and Owen, A. (1997). Valuation of Mortgage-Backed Securities Using Brownian Bridges to Reduce Effective Dimension., The Journal of Computational Finance 1 27–46.
  • Conway, J. H. and Sloane, N. J. A. (1999)., Sphere Packings, Lattices and Groups, 3rd ed. Grundlehren der Mathematischen Wissenschaften 290. Springer-Verlag, New York.
  • Cranley, R. and Patterson, T. N. L. (1976). Randomization of Number Theoretic Methods for Multiple Integration., SIAM Journal on Numerical Analysis 13 904–914.
  • Dick, J., Sloan, I. H., Wang, X. and Wozniakowski, H. (2006). Good Lattice Rules in Weighted Korobov Spaces with General Weights., Numerische Mathematik 103 63–97.
  • Elmaghraby, S. (1977)., Activity Networks. Wiley, New York.
  • Glasserman, P. (2004)., Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York.
  • Hickernell, F. J. (2002). Obtaining, O(N2+ε) Convergence for Lattice Quadrature Rules. In Monte Carlo and Quasi-Monte Carlo Methods 2000 ( K.-T. Fang, F. J. Hickernell and H. Niederreiter, eds.) 274–289. Springer-Verlag, Berlin.
  • Hull, J. C. (2000)., Options, Futures, and Other Derivative Securities, fourth ed. Prentice-Hall, Englewood-Cliff, N.J.
  • Kuo, F. Y. and Joe, S. (2002). Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points., Journal of Complexity 18 943–976.
  • L’Ecuyer, P. (2009). Quasi-Monte Carlo Methods with Applications in Finance., Finance and Stochastics 13 307–349.
  • L’Ecuyer, P. and Lemieux, C. (2000). Variance Reduction via Lattice Rules., Management Science 46 1214–1235.
  • L’Ecuyer, P. and Lemieux, C. (2002). Recent Advances in Randomized Quasi-Monte Carlo Methods. In, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications ( M. Dror, P. L’Ecuyer and F. Szidarovszky, eds.) 419–474. Kluwer Academic, Boston.
  • L’Ecuyer, P. and Tuffin, B. (2009). On the Error Distribution for Randomly-Shifted Lattice Rules. In, Proceedings of the 2009 Winter Simulation Conference. IEEE Press To appear.
  • Lemieux, C. (2009)., Monte Carlo and Quasi-Monte Carlo Sampling. Springer-Verlag, New York, NY.
  • Liu, R. and Owen, A. B. (2006). Estimating Mean Dimensionality of Analysis of Variance Decompositions., Journal of the American Statistical Association 101 712–721.
  • Loh, W.-L. (2003). On the Asymptotic Distribution of Scramble Nets Quadratures., Annals of Statistics 31 1282–1324.
  • Loh, W.-L. (2005). On the Asymptotic Distribution of Some Randomized Quadrature Rules. In, Stein’s Method and Applications, ( C. Stein, A. D. Barbour and L. H. Y. Chen, eds.) 5 209–222. World Scientific.
  • Niederreiter, H. (1992)., Random Number Generation and Quasi-Monte Carlo Methods. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics 63. SIAM, Philadelphia, PA.
  • Owen, A. B. (1992). A Central Limit Theorem for Latin Hypercube Sampling., Journal of the Royal Statistical Society B 54 541–551.
  • Owen, A. B. (1997). Scrambled Net Variance for Integrals of Smooth Functions., Annals of Statistics 25 1541–1562.
  • Owen, A. B. (1998). Latin Supercube Sampling for Very High-Dimensional Simulations., ACM Transactions on Modeling and Computer Simulation 8 71–102.
  • Owen, A. B. (2003). Variance with Alternative Scramblings of Digital Nets., ACM Transactions on Modeling and Computer Simulation 13 363–378.
  • Petrov, V. V. (1995)., Limit Theorems of Probability Theory. Oxford University Press, Oxford, U.K.
  • Sinescu, V. and L’Ecuyer, P. (2010). Existence and Contruction of Shifted Lattice Rules with an Arbitrary Number of Points and Bounded Weighted Star Discrepancy for General Weights. Submitted for, publication.
  • Sloan, I. H. and Joe, S. (1994)., Lattice Methods for Multiple Integration. Clarendon Press, Oxford.
  • Sloan, I. H. and Rezstov, A. (2002). Component-by-Component Construction of Good Lattice Rules., Mathematics of Computation 71 262–273.
  • Sobol’, I. M. and Myshetskaya, E. E. (2007). Monte Carlo Estimators for Small Sensitivity Indices., Monte Carlo Methods and Applications 13 455–465.
  • Stein, M. (1987). Large Sample Properties of Simulations Using Latin Hypercube Sampling., Technometrics 29 143–151.
  • Tuffin, B. (1998). Variance Reduction Order Using Good Lattice Points in Monte Carlo Methods., Computing 61 371–378.