Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. (English) Zbl 1200.62010

Stein’s method has been suggested to asses the distance between univariate random variables and normal distributions. It makes use of the method of exchangeable pairs [see C. Stein, Inst. Math. Stat. Lect. Notes - Monogr. Ser. 7 (1986; Zbl 0721.60016); A. Röllin, Stat. Probab. Lett. 78, No. 13, 1800–1806 (2008; Zbl 1147.62017); and S. Chatterjee and E. Meckes, ALEA, Lat. Am. J. Probab. Math. Stat. 4, 257–283, electronic only (2008; Zbl 1162.60310)]. The paper establishes a multivariate exchangeable pairs approach enabling to asses distances to potentially singular multivariate normal distributions. An embedding method is suggested allowing for a normal approximation even when the corresponding statistics are inappropriate for Stein’s exchangeable pairs approach. Examples of runs on the line and double-indexed permutation statistics illustrate the method.


62E17 Approximations to statistical distributions (nonasymptotic)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60F05 Central limit and other weak theorems
60C05 Combinatorial probability
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[1] Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications . Wiley, New York. · Zbl 0991.62087
[2] Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297-322. · Zbl 0665.60008
[3] Barbour, A. D. and Chen, L. H. Y. (2005). The permutation distribution of matrix correlation statistics. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 223-245. Singapore Univ. Press, Singapore.
[4] Bhattacharya, R. N. and Holmes, S. (2007). An exposition of Götze’s paper.
[5] Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379-386. · Zbl 0563.60026
[6] Bolthausen, E. and Götze, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21 1692-1710. · Zbl 0798.62023
[7] Chatterjee, S., Fulman, J. and Röllin, A. (2006). Exponential approximation by exchangeable pairs and spectral graph theory. Preprint. Available at www.arxiv.org/math.PR/0605552.
[8] Chatterjee, S. and Meckes, E. (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 257-283. · Zbl 1162.60310
[9] Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64-106 (electronic). · Zbl 1189.60072
[10] Fulman, J. (2004). Stein’s method and non-reversible Markov chains. In Stein’s Method : Expository Lectures and Applications (P. Diaconis and S. Holmes, eds.) 69-77. IMS, Beachwood, OH.
[11] Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics . Springer, New York. · Zbl 0983.62075
[12] Goldstein, L. and Reinert, G. (2005). Zero biasing in one and higher dimensions, and applications. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 1-18. Singapore Univ. Press, Singapore.
[13] Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33 1-17. JSTOR: · Zbl 0845.60023
[14] Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724-739. · Zbl 0729.62051
[15] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Statist. 22 558-566. · Zbl 0044.13702
[16] Lemeire, F. (1975). Bounds for condition numbers of triangular and trapezoid matrices. Nordisk Tidskr. Informationsbehandling ( BIT ) 15 58-64. · Zbl 0325.65018
[17] Loh, W.-L. (2008). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 1983-2023. · Zbl 1143.62044
[18] Mann, H. B. and Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Statist. 18 50-60. · Zbl 0041.26103
[19] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis . Academic Press, London. · Zbl 0432.62029
[20] Raič, M. (2004). A multivariate CLT for decomposable random vectors with finite second moments. J. Theoret. Probab. 17 573-603. · Zbl 1059.62050
[21] Reinert, G. (2005). Three general approaches to Stein’s method. In An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 183-221. Singapore Univ. Press, Singapore.
[22] Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with n - 1/2 log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333-350. · Zbl 0859.60019
[23] Rinott, Y. and Rotar, V. (1997). On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U -statistics. Ann. Appl. Probab. 7 1080-1105. · Zbl 0890.60019
[24] Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab. 17 1596-1614. · Zbl 1143.60020
[25] Röllin, A. (2008). A note on the exchangeability condition in Stein’s method. Statist. Probab. Lett. 78 1800-1806. · Zbl 1147.62017
[26] Rotar, V. (1997). Probability Theory . World Scientific, River Edge, NJ. · Zbl 0871.60002
[27] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. Vol. II : Probability Theory . Univ. California Press, Berkeley, CA. · Zbl 0278.60026
[28] Stein, C. (1986). Approximate Computation of Expectations . IMS, Hayward, CA. · Zbl 0721.60016
[29] Zhao, L., Bai, Z., Chao, C.-C. and Liang, W.-Q. (1997). Error bound in a central limit theorem of double-indexed permutation statistics. Ann. Statist. 25 2210-2227. · Zbl 0897.60024
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