The Annals of Statistics

On the asymptotic distribution of scrambled net quadrature

Wei-Liem Loh

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Abstract

Recently, in a series of articles, Owen proposed the use of scrambled (t.m.s) nets and (t.s) sequences in high-dimensional numerical integration. These scrambled nets and sequences achieve the superior accuracy of equidistribution methods while allowing for the simpler error estimation techniques of Monte Carlo methods. The main aim of this article is to use Stein's method to study the asymptotic distribution of the scrambled (0.m.s) net integral estimate. In particular, it is shown that, for suitably smooth integrands on the s-dimensional unit hypercube, the estimate has an asymptotic normal distribution.

Article information

Source
Ann. Statist., Volume 31, Number 4 (2003), 1282-1324.

Dates
First available in Project Euclid: 31 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1059655914

Digital Object Identifier
doi:10.1214/aos/1059655914

Mathematical Reviews number (MathSciNet)
MR2001651

Zentralblatt MATH identifier
1105.62304

Subjects
Primary: 62E05
Secondary: 62D05: Sampling theory, sample surveys 65C05: Monte Carlo methods

Keywords
Asymptotic normality computer experiment numerical integration quasi-Monte Carlo scrambled net Stein's method

Citation

Loh, Wei-Liem. On the asymptotic distribution of scrambled net quadrature. Ann. Statist. 31 (2003), no. 4, 1282--1324. doi:10.1214/aos/1059655914. https://projecteuclid.org/euclid.aos/1059655914


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References

  • BOLTHAUSEN, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379-386.
  • BOLTHAUSEN, E. and GÖTZE, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21 1692-1710.
  • DAVIS, P. J. and RABINOWITZ, P. (1984). Methods of Numerical Integration, 2nd ed. Academic Press, Orlando, FL.
  • EVANS, M. and SWARTZ, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford Univ. Press.
  • HICKERNELL, F. J. and YUE, R.-X. (2000). The mean square discrepancy of scrambled (t, s)-sequences. SIAM J. Numer. Anal. 38 1089-1112.
  • HO, S. T. and CHEN, L. H. Y. (1978). An Lp bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231-249.
  • HONG, H. S., HICKERNELL, F. J. and WEI, G. (2001). The distribution of the discrepancy of scrambled digital (t, m, s)-nets. Preprint.
  • LOH, W.-L. (1996a). A combinatorial central limit theorem for randomized orthogonal array sampling designs. Ann. Statist. 24 1209-1224.
  • LOH, W.-L. (1996b). On Latin hy percube sampling. Ann. Statist. 24 2058-2080.
  • MCKAY, M. D., BECKMAN, R. J. and CONOVER, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 239-245.
  • NIEDERREITER, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia.
  • NIEDERREITER, H. and XING, C. (1998). Nets, (t, s)-sequences, and algebraic geometry. Random and Quasi-Random Point Sets. Lecture Notes in Statist. 138 267-302. Springer, New York.
  • OWEN, A. B. (1992a). Orthogonal array s for computer experiments, integration and visualization. Statist. Sinica 2 439-452.
  • OWEN, A. B. (1992b). A central limit theorem for Latin hy percube sampling. J. R. Statist. Soc. Ser. B 54 541-551.
  • OWEN, A. B. (1995). Randomly permuted (t, m, s)-nets and (t, s)-sequences. Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statist. 106 299-317. Springer, New York.
  • OWEN, A. B. (1997a). Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34 1884-1910.
  • OWEN, A. B. (1997b). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541-1562.
  • OWEN, A. B. (1998). Scrambling Sobol' and Niederreiter-Xing points. J. Complexity 14 466-489.
  • SHORACK, G. R. (2000). Probability for Statisticians. Springer, New York.
  • STEIN, C. M. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 2 583-602. Univ. California Press, Berkeley.
  • STEIN, C. M. (1986). Approximate Computation of Expectations. IMS, Hay ward, CA.
  • STEIN, M. L. (1987). Large sample properties of simulations using Latin hy percube sampling. Technometrics 29 143-151.
  • TANG, B. (1993). Orthogonal array-based Latin hy percubes. J. Amer. Statist. Assoc. 88 1392-1397.
  • YUE, R.-X. (1999). Variance of quadrature over scrambled unions of nets. Statist. Sinica 9 451-473.
  • YUE, R.-X. and MAO, S.-S. (1999). On the variance of quadrature over scrambled nets and sequences. Statist. Probab. Lett. 44 267-280.