Asymptotic distribution of the largest off-diagonal entry of correlation matrices
Author: Wang Zhou
Journal: Trans. Amer. Math. Soc. 359 (2007), 5345-5363
MSC (2000): Primary 60F05, 62G20, 62H10
Published electronically: May 11, 2007
MathSciNet review: 2327033
Abstract: Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as . A test statistic is chosen to be , where is the sample correlation coefficient between the -th coordinate and the -th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of is an extreme distribution of type , by using the Chen-Stein Poisson approximation method and the moderate deviations for sample correlation coefficients. As a statistically more relevant result, a limit distribution for , where is Spearman's rank correlation coefficient between the -th coordinate and the -th coordinate of the population, is derived.
- 1. R. Arratia, L. Goldstein, and L. Gordon, Two moments suffice for Poisson approximations: the Chen-Stein method, Ann. Probab. 17 (1989), no. 1, 9–25. MR 972770
- 2. A. D. Barbour and G. K. Eagleson, Poisson convergence for dissociated statistics, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 397–402. MR 790624
- 3. Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507-521.
- 4. Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests. Academic Press, New York.
- 5. Myles Hollander and Douglas A. Wolfe, Nonparametric statistical methods, 2nd ed., Wiley Series in Probability and Statistics: Texts and References Section, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1666064
- 6. Harold Hotelling, New light on the correlation coefficient and its transforms, J. Roy. Statist. Soc. Ser. B. 15 (1953), 193–225; discussion, 225–232. MR 0060794
- 7. Tiefeng Jiang, The asymptotic distributions of the largest entries of sample correlation matrices, Ann. Appl. Probab. 14 (2004), no. 2, 865–880. MR 2052906, https://doi.org/10.1214/105051604000000143
- 8. Bing-Yi Jing, Qi-Man Shao, and Qiying Wang, Self-normalized Cramér-type large deviations for independent random variables, Ann. Probab. 31 (2003), no. 4, 2167–2215. MR 2016616, https://doi.org/10.1214/aop/1068646382
- 9. Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295–327. MR 1863961, https://doi.org/10.1214/aos/1009210544
- 10. Qi-Man Shao, Self-normalized large deviations, Ann. Probab. 25 (1997), no. 1, 285–328. MR 1428510, https://doi.org/10.1214/aop/1024404289
- 11. Qi-Man Shao, A Cramér type large deviation result for Student’s 𝑡-statistic, J. Theoret. Probab. 12 (1999), no. 2, 385–398. MR 1684750, https://doi.org/10.1023/A:1021626127372
- 12. Munsup Seoh, Stefan S. Ralescu, and Madan L. Puri, Cramér type large deviations for generalized rank statistics, Ann. Probab. 13 (1985), no. 1, 115–125. MR 770632
- 13. Spearman, C. (1904). The proof and measurement of association between two things. Amer. J. Psychol., 15, 72-101.
- 14. Qiying Wang and Bing-Yi Jing, An exponential nonuniform Berry-Esseen bound for self-normalized sums, Ann. Probab. 27 (1999), no. 4, 2068–2088. MR 1742902, https://doi.org/10.1214/aop/1022677562
- Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximation: the Chen-Stein method. Ann. Prob., 17, 9-25. MR 972770 (90b:60021)
- Barbour, A. and Eagleson, G. (1984). Poisson convergence for dissociated statistics. J. R. Statist. Soc. B, 46, 397-402. MR 790624 (86k:60033)
- Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507-521.
- Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests. Academic Press, New York.
- Hollander, M. and Wolfe, D. A. (1999). Nonparametric Statistical Methods. Wiley, New York. MR 1666064 (99m:62004)
- Hotelling, H. (1953). New light on the correlation coefficient and its transforms. (with discussion) J. Roy. Statist. Soc. Ser. B., 15, 193-232. MR 0060794 (15:728d)
- Jiang, T. (2004). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Prob., 14, 865-880. MR 2052906 (2005b:60053)
- Jing, B.-Y., Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab., 31, 2167-2215. MR 2016616 (2004k:60069)
- Johnstone, I. (2001). On the distribution of the largest eigenvalue in principal component analysis. Ann. Stat., 29, 295-327. MR 1863961 (2002i:62115)
- Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab., 25, 285-328. MR 1428510 (98b:60056)
- Shao, Q.-M. (1999). A Cramér type large deviation result for Student's -statistic. J. Theoret. Probab., 12, 385-398. MR 1684750 (2000d:60046)
- Seoh, M., Ralescu, S. and Puri, M. L. (1985). Cramér type large deviations for generalized rank statistics. Ann. Prob., 13, 115-125. MR 770632 (86k:62077)
- Spearman, C. (1904). The proof and measurement of association between two things. Amer. J. Psychol., 15, 72-101.
- Wang, Q. Y. and Jing, B.-Y. (1999). An exponential non-uniform Berry-Esséen bound for self-normalized sums. Ann. Probab., 27, 2068-2088. MR 1742902 (2001c:60045)
Affiliation: Department of Statistics and Applied Probability, National University of Singapore, Singapore, 117546
Keywords: Sample correlation matrices, Spearman's rank correlation matrices, Chen-Stein method, moderate deviations.
Received by editor(s): April 25, 2005
Received by editor(s) in revised form: September 5, 2005
Published electronically: May 11, 2007
Additional Notes: The author was supported in part by grants R-155-000-035-112 and R-155-050-055-133/101 at the National University of Singapore
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.