The Annals of Statistics

A central limit theorem for general orthogonal array based space-filling designs

Xu He and Peter Z. G. Qian

Full-text: Open access


Orthogonal array based space-filling designs (Owen [Statist. Sinica 2 (1992a) 439–452]; Tang [J. Amer. Statist. Assoc. 88 (1993) 1392–1397]) have become popular in computer experiments, numerical integration, stochastic optimization and uncertainty quantification. As improvements of ordinary Latin hypercube designs, these designs achieve stratification in multi-dimensions. If the underlying orthogonal array has strength $t$, such designs achieve uniformity up to $t$ dimensions. Existing central limit theorems are limited to these designs with only two-dimensional stratification based on strength two orthogonal arrays. We develop a new central limit theorem for these designs that possess stratification in arbitrary multi-dimensions associated with orthogonal arrays of general strength. This result is useful for building confidence statements for such designs in various statistical applications.

Article information

Ann. Statist., Volume 42, Number 5 (2014), 1725-1750.

First available in Project Euclid: 11 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62K99: None of the above, but in this section 05B15: Orthogonal arrays, Latin squares, Room squares

Computer experiment design of experiment method of moment numerical integration uncertainty quantification


He, Xu; Qian, Peter Z. G. A central limit theorem for general orthogonal array based space-filling designs. Ann. Statist. 42 (2014), no. 5, 1725--1750. doi:10.1214/14-AOS1231.

Export citation


  • Birge, J. R. and Louveaux, F. (2011). Introduction to Stochastic Programming, 2nd ed. Springer, New York.
  • Branin, F. H. Jr. (1972). Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. IBM J. Res. Develop. 16 504–522.
  • Cox, D. D., Park, J.-S. and Singer, C. E. (2001). A statistical method for tuning a computer code to a data base. Comput. Statist. Data Anal. 37 77–92.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
  • 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. (2008). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 1983–2023.
  • 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.
  • Owen, D. B. (1980). A table of normal integrals. Comm. Statist. B—Simulation Comput. 9 389–419.
  • 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. R. Stat. Soc. Ser. B Stat. Methodol. 54 541–551.
  • Owen, A. (1994). Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. Ann. Statist. 22 930–945.
  • Owen, A. B. (1997). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541–1562.
  • Patterson, H. D. (1954). The errors of lattice sampling. J. R. Stat. Soc. Ser. B Stat. Methodol. 16 140–149.
  • Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. MPS/SIAM Series on Optimization 9. SIAM, Philadelphia, PA.
  • Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc. 88 1392–1397.
  • Tang, Q. and Qian, P. Z. G. (2010). Enhancing the sample average approximation method with $U$ designs. Biometrika 97 947–960.
  • Xiu, D. (2010). Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton Univ. Press, Princeton, NJ.