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Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. (English) Zbl 0807.62059
Summary: Randomized orthogonal arrays provide good sets of input points for exploration of computer programs and for Monte Carlo integration. H. D. Patterson [J. R. Stat. Soc., Ser. B 16, 140-149 (1954; Zbl 0056.381)] gave a formula for the randomization variance of the sample mean of a function evaluated at the points of an orthogonal array. That formula is incorrect for most of the arrays that are practical for computer experiments. In this paper we correct Patterson’s formula.
We also remark on a defect, related to coincidences, in some orthogonal arrays. These are arrays of the form $$OA( 2q^ 2, 2q+1,q,2)$$, where $$q$$ is a prime power, obtained by constructions due to R. C. Bose and K. A. Bush [Ann. Math. Stat. 23, 508-524 (1952; Zbl 0048.008)] and to S. Addelman and O. Kempthorne [ibid. 32, 1167-1176 (1961; Zbl 0107.360)]. We conjecture that subarrays of the form $$OA (2q^ 2, 2q,q ,2)$$ may be constructed to avoid this defect.

##### MSC:
 62K15 Factorial statistical designs 65C05 Monte Carlo methods 62K99 Design of statistical experiments 05B15 Orthogonal arrays, Latin squares, Room squares 65D30 Numerical integration
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