The Annals of Statistics

A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments

Wei-Liem Loh

Full-text: Open access

Abstract

Let f : [0, 1)d→ℝ be an integrable function. An objective of many computer experiments is to estimate [0, 1)df(x) dx by evaluating f at a finite number of points in [0, 1)d. There is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays. This article proves a multivariate central limit theorem for a class of randomized orthogonal array sampling designs [Owen Statist. Sinica 2 (1992a) 439–452] as well as for a class of OA-based Latin hypercubes [Tang J. Amer. Statist. Assoc. 81 (1993) 1392–1397].

Article information

Source
Ann. Statist. Volume 36, Number 4 (2008), 1983-2023.

Dates
First available in Project Euclid: 16 July 2008

Permanent link to this document
http://projecteuclid.org/euclid.aos/1216237306

Digital Object Identifier
doi:10.1214/07-AOS530

Mathematical Reviews number (MathSciNet)
MR2435462

Zentralblatt MATH identifier
1143.62044

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems 65C05: Monte Carlo methods

Keywords
Computer experiment multivariate central limit theorem numerical integration OA-based Latin hypercube randomized orthogonal array Stein’s method

Citation

Loh, Wei-Liem. A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 (2008), no. 4, 1983--2023. doi:10.1214/07-AOS530. http://projecteuclid.org/euclid.aos/1216237306.


Export citation

References

  • 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.
  • Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724–739.
  • Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
  • Loh, W. L. (1996). A combinatorial central limit theorem for randomized orthogonal array sampling designs. Ann. Statist. 24 1209–1224.
  • Loh, W. L. (2007). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Available at http://arxiv.org/abs/0708.0656v1.
  • McKay, M. D., Conover, W. J. and Beckman, R. J. (1979). A comparison of three methods for selecting values of output variables in the analysis of output from a computer code. Technometrics 21 239–245.
  • Owen, A. B. (1992a). Orthogonal arrays for computer experiments, integration and visualization. Statist. Sinica 2 439–452.
  • Owen, A. B. (1992b). A central limit theorem for Latin hypercube sampling. J. Roy. Statist. Soc. Ser. B 54 541–551.
  • Owen, A. B. (1994). Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. Ann. Statist. 22 930–945.
  • 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.
  • Raghavarao, D. (1971). Constructions and Combinatorial Problems in Design of Experiments. Wiley, New York.
  • Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409–423.
  • Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments. 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 Symp. Math. Statist. Probab. 2 583–602. Univ. California Press, Berkeley.
  • Stein, C. M. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.
  • Stein, M. L. (1987). Large sample properties of simulations using Latin hypercube sampling. Technometrics 29 143–151.
  • Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc. 88 1392–1397.