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A combinatorial central limit theorem for randomized orthogonal array sampling designs. (English) Zbl 0869.62018

Let \(X\) be a random vector uniformly distributed on \([0,1]^3\) and \(f\) be a measurable function from \([0,1]^3\) to the real line. An objective of many computer experiments is to estimate \(\mu= E(f \circ x)\) by computing \(f\) at a fixed number of points. A. B. Owen [ibid. 22, No. 2, 930-945 (1994; Zbl 0807.62059); Stat. Sinica 2, No. 2, 439-452 (1992; Zbl 0822.62064)] and B. Tang [J. Am. Stat. Assoc. 88, No. 424, 1392-1397 (1993; Zbl 0792.62066)] independently suggested the use of randomized orthogonal arrays in sampling designs for computer experiments. The main attraction of these designs is that they, in contrast to simple random sampling, stratify on all \(t\)-variate margins simultaneously.
C. M. Stein’s method [see “Approximate computation of expectations.” (1986; Zbl 0721.60016)] is used by the author to prove asymptotic normality of \(\widehat\mu\) which is the usual average of the orthogonal array-based sample of size \(q^2\) of \(f\) values. In fact, he shows that under the finiteness of \(r\)-th moments the corresponding error bound is of the order \(O(q^{-(r-2)/(2r-2)})\), where \(r\) is an even integer greater than or equal to 4.

MSC:

62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
62K10 Statistical block designs
62K99 Design of statistical experiments
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[1] HO, S. T. and CHEN, L. H. Y. 1978. An L bound for the remainder in a combinatorial central p limit theorem. Ann. Probab. 6 231 249. Z. · Zbl 0375.60028 · doi:10.1214/aop/1176995570
[2] 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. Z. JSTOR: · Zbl 0415.62011 · doi:10.2307/1268522
[3] OWEN, A. B. 1992. Orthogonal array s for computer experiments, integration and visualization. Statist. Sinica 2 439 452. Z. · Zbl 0822.62064
[4] OWEN, A. B. 1994. Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal array s. Ann. Statist. 22 930 945. · Zbl 0807.62059 · doi:10.1214/aos/1176325504
[5] RAGHAVARAO, D. 1971. Constructions and Combinatorial Problems in Design of Experiments. Wiley, New York. Z. · Zbl 0222.62036
[6] 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. Z. · Zbl 0278.60026
[7] STEIN, C. M. 1986. Approximate Computation of Expectations. IMS, Hay ward, CA. Z. · Zbl 0721.60016
[8] STEIN, M. L. 1987. Large sample properties of simulations using Latin hy percube sampling. Technometrics 29 143 151. Z. JSTOR: · Zbl 0627.62010 · doi:10.2307/1269769
[9] TANG, B. 1993. Orthogonal array-based Latin hy percubes. J. Amer. Statist. Assoc. 88 1392 1397. JSTOR: · Zbl 0792.62066 · doi:10.2307/2291282
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