Loh, Wei-Liem A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. (English) Zbl 1143.62044 Ann. Stat. 36, No. 4, 1983-2023 (2008). Summary: Let \(f:[0,1)^d\to\mathbb R\) be an integrable function. An objective of many computer experiments is to estimate \(\int_{[0, 1)^d} f(x)\,dx\) 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 [A. B. Owen, Stat. Sin. 2, No. 2, 439–452 (1992; Zbl 0822.62064); Corrigendumm ibid. 3, No. 1, 260 (1992)] as well as for a class of OA-based Latin hypercubes [B. Tang, J. Am. Stat. Assoc. 88, No. 424, 1392–1397 (1993; Zbl 0792.62066)]. Cited in 8 Documents MSC: 62K99 Design of statistical experiments 60F05 Central limit and other weak theorems 65C60 Computational problems in statistics (MSC2010) 05B15 Orthogonal arrays, Latin squares, Room squares 62E20 Asymptotic distribution theory in statistics 65D30 Numerical integration 62J10 Analysis of variance and covariance (ANOVA) Keywords:numerical integration; OA-based Latin hypercube; Stein’s method; ANOVA decomposition Citations:Zbl 0822.62064; Zbl 0792.62066 PDFBibTeX XMLCite \textit{W.-L. Loh}, Ann. Stat. 36, No. 4, 1983--2023 (2008; Zbl 1143.62044) Full Text: DOI arXiv References: [1] Bolthausen, E. and Götze, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21 1692-1710. · Zbl 0798.62023 · doi:10.1214/aos/1176349393 [2] Davis, P. J. and Rabinowitz, P. (1984). Methods of Numerical Integration , 2nd ed. Academic Press, Orlando. · Zbl 0537.65020 [3] Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724-739. · Zbl 0729.62051 · doi:10.1214/aop/1176990448 [4] Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays : Theory and Applications . Springer, New York. · Zbl 0935.05001 [5] Loh, W. L. (1996). A combinatorial central limit theorem for randomized orthogonal array sampling designs. Ann. Statist. 24 1209-1224. · Zbl 0869.62018 · doi:10.1214/aos/1032526964 [6] 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. · Zbl 1143.62044 · doi:10.1214/07-AOS530 [7] 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. JSTOR: · Zbl 0415.62011 · doi:10.2307/1268522 [8] Owen, A. B. (1992a). Orthogonal arrays for computer experiments, integration and visualization. Statist. Sinica 2 439-452. · Zbl 0822.62064 [9] Owen, A. B. (1992b). A central limit theorem for Latin hypercube sampling. J. Roy. Statist. Soc. Ser. B 54 541-551. JSTOR: · Zbl 0776.62041 [10] Owen, A. B. (1994). Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. Ann. Statist. 22 930-945. · Zbl 0807.62059 · doi:10.1214/aos/1176325504 [11] Owen, A. B. (1997a). Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34 1884-1910. JSTOR: · Zbl 0890.65023 · doi:10.1137/S0036142994277468 [12] Owen, A. B. (1997b). Scrambled net variance for integrals of smooth functions. Ann. Statist. 25 1541-1562. · Zbl 0886.65018 · doi:10.1214/aos/1031594731 [13] Raghavarao, D. (1971). Constructions and Combinatorial Problems in Design of Experiments . Wiley, New York. · Zbl 0222.62036 [14] Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409-423. · Zbl 0955.62619 · doi:10.1214/ss/1177012413 [15] Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments . Springer, New York. · Zbl 1041.62068 [16] 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. · Zbl 0278.60026 [17] Stein, C. M. (1986). Approximate Computation of Expectations . IMS, Hayward, CA. · Zbl 0721.60016 [18] Stein, M. L. (1987). Large sample properties of simulations using Latin hypercube sampling. Technometrics 29 143-151. JSTOR: · Zbl 0627.62010 · doi:10.2307/1269769 [19] Tang, B. (1993). Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc. 88 1392-1397. JSTOR: · Zbl 0792.62066 · doi:10.2307/2291282 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.