Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition
Gesine Reinert and Adrian Röllin
Source: Ann. Probab. Volume 37, Number 6
(2009), 2150-2173.
Abstract
In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein’s method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein’s exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380785
Digital Object Identifier: doi:10.1214/09-AOP467
Mathematical Reviews number (MathSciNet): MR2573554
Zentralblatt MATH identifier: 1200.62010
References
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1882476
Zentralblatt MATH: 0991.62087
Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297–322.
Mathematical Reviews (MathSciNet): MR1035659
Zentralblatt MATH: 0665.60008
Digital Object Identifier: doi:10.1007/BF01197887
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.
Mathematical Reviews (MathSciNet): MR2205339
Bhattacharya, R. N. and Holmes, S. (2007). An exposition of Götze’s paper. Preprint.
Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379–386.
Mathematical Reviews (MathSciNet): MR751577
Bolthausen, E. and Götze, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21 1692–1710.
Mathematical Reviews (MathSciNet): MR1245764
Zentralblatt MATH: 0798.62023
Digital Object Identifier: doi:10.1214/aos/1176349393
Project Euclid: euclid.aos/1176349393
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.
Chatterjee, S. and Meckes, E. (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 257–283.
Mathematical Reviews (MathSciNet): MR2453473
Zentralblatt MATH: 1162.60310
Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64–106 (electronic).
Mathematical Reviews (MathSciNet): MR2121796
Digital Object Identifier: doi:10.1214/154957805100000096
Project Euclid: euclid.ps/1109608868
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.
Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.
Mathematical Reviews (MathSciNet): MR1869112
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.
Mathematical Reviews (MathSciNet): MR2201883
Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33 1–17.
Mathematical Reviews (MathSciNet): MR1371949
Zentralblatt MATH: 0845.60023
Digital Object Identifier: doi:10.2307/3215259
JSTOR: links.jstor.org
Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724–739.
Mathematical Reviews (MathSciNet): MR1106283
Zentralblatt MATH: 0729.62051
Digital Object Identifier: doi:10.1214/aop/1176990448
Project Euclid: euclid.aop/1176990448
Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Statist. 22 558–566.
Mathematical Reviews (MathSciNet): MR44058
Digital Object Identifier: doi:10.1214/aoms/1177729545
Project Euclid: euclid.aoms/1177729545
Lemeire, F. (1975). Bounds for condition numbers of triangular and trapezoid matrices. Nordisk Tidskr. Informationsbehandling (BIT) 15 58–64.
Mathematical Reviews (MathSciNet): MR501837
Loh, W.-L. (2008). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 1983–2023.
Mathematical Reviews (MathSciNet): MR2435462
Digital Object Identifier: doi:10.1214/07-AOS530
Project Euclid: euclid.aos/1216237306
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.
Mathematical Reviews (MathSciNet): MR22058
Digital Object Identifier: doi:10.1214/aoms/1177730491
Project Euclid: euclid.aoms/1177730491
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
Mathematical Reviews (MathSciNet): MR560319
Raič, M. (2004). A multivariate CLT for decomposable random vectors with finite second moments. J. Theoret. Probab. 17 573–603.
Mathematical Reviews (MathSciNet): MR2091552
Digital Object Identifier: doi:10.1023/B:JOTP.0000040290.44087.68
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.
Mathematical Reviews (MathSciNet): MR2235451
Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with n−1/2log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333–350.
Mathematical Reviews (MathSciNet): MR1379533
Digital Object Identifier: doi:10.1006/jmva.1996.0017
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.
Mathematical Reviews (MathSciNet): MR1484798
Digital Object Identifier: doi:10.1214/aoap/1043862425
Project Euclid: euclid.aoap/1043862425
Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab. 17 1596–1614.
Mathematical Reviews (MathSciNet): MR2358635
Digital Object Identifier: doi:10.1214/105051607000000258
Project Euclid: euclid.aoap/1191419177
Röllin, A. (2008). A note on the exchangeability condition in Stein’s method. Statist. Probab. Lett. 78 1800–1806.
Mathematical Reviews (MathSciNet): MR2453918
Rotar, V. (1997). Probability Theory. World Scientific, River Edge, NJ.
Mathematical Reviews (MathSciNet): MR1641490
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.
Mathematical Reviews (MathSciNet): MR402873
Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet): MR882007
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.
Mathematical Reviews (MathSciNet): MR1474091
Digital Object Identifier: doi:10.1214/aos/1069362395
Project Euclid: euclid.aos/1069362395
The Annals of Probability
