The Annals of Probability

Largest entries of sample correlation matrices from equi-correlated normal populations

Jianqing Fan and Tiefeng Jiang

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

Abstract

The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient $\rho >0$ and both the population dimension $p$ and the sample size $n$ tend to infinity with $\log p=o(n^{\frac{1}{3}})$. As $0<\rho <1$, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as $0<\rho <1/2$. This differs substantially from a well-known result for the independent case where $\rho =0$, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of $\rho $ where the transition occurs. If $\rho $ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen–Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.

Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3321-3374.

Dates
Received: September 2017
Revised: January 2019
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731453

Digital Object Identifier
doi:10.1214/19-AOP1341

Mathematical Reviews number (MathSciNet)
MR4021253

Subjects
Primary: 62H10: Distribution of statistics 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Keywords
Maximum sample correlation phase transition multivariate normal distribution Gumbel distribution Chen–Stein Poisson approximation

Citation

Fan, Jianqing; Jiang, Tiefeng. Largest entries of sample correlation matrices from equi-correlated normal populations. Ann. Probab. 47 (2019), no. 5, 3321--3374. doi:10.1214/19-AOP1341. https://projecteuclid.org/euclid.aop/1571731453


Export citation

References

  • [1] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
    Zentralblatt MATH: 0675.60017
    Digital Object Identifier: doi:10.1214/aop/1176991491
    Project Euclid: euclid.aop/1176991491
  • [2] Cai, T., Fan, J. and Jiang, T. (2013). Distributions of angles in random packing on spheres. J. Mach. Learn. Res. 14 1837–1864.
    Zentralblatt MATH: 1318.60017
  • [3] Cai, T. T. and Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 1496–1525.
    Zentralblatt MATH: 1220.62066
    Digital Object Identifier: doi:10.1214/11-AOS879
    Project Euclid: euclid.aos/1305292044
  • [4] Cai, T. T. and Jiang, T. (2012). Phase transition in limiting distributions of coherence of high-dimensional random matrices. J. Multivariate Anal. 107 24–39.
    Zentralblatt MATH: 1352.60006
    Digital Object Identifier: doi:10.1016/j.jmva.2011.11.008
  • [5] Chen, L. H. Y., Fang, X. and Shao, Q.-M. (2013). From Stein identities to moderate deviations. Ann. Probab. 41 262–293.
    Zentralblatt MATH: 1275.60029
    Digital Object Identifier: doi:10.1214/12-AOP746
    Project Euclid: euclid.aop/1358951987
  • [6] Chernozhukov, V., Chetverikov, D. and Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. Ann. Statist. 41 2786–2819.
    Zentralblatt MATH: 1292.62030
    Digital Object Identifier: doi:10.1214/13-AOS1161
    Project Euclid: euclid.aos/1387313390
  • [7] Eaton, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
    Zentralblatt MATH: 0587.62097
  • [8] Fan, J., Shao, Q.-M. and Zhou, W.-X. (2018). Are discoveries spurious? Distributions of maximum spurious correlations and their applications. Ann. Statist. 46 989–1017.
    Zentralblatt MATH: 1402.62097
    Digital Object Identifier: doi:10.1214/17-AOS1575
    Project Euclid: euclid.aos/1525313073
  • [9] Jiang, T. (2004). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880.
    Zentralblatt MATH: 1047.60014
    Digital Object Identifier: doi:10.1214/105051604000000143
    Project Euclid: euclid.aoap/1082737115
  • [10] Jiang, T. (2004). The limiting distributions of eigenvalues of sample correlation matrices. Sankhyā 66 35–48.
    Zentralblatt MATH: 1193.62018
  • [11] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer, New York.
  • [12] Li, D., Liu, W.-D. and Rosalsky, A. (2010). Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Related Fields 148 5–35.
    Zentralblatt MATH: 1210.62010
    Digital Object Identifier: doi:10.1007/s00440-009-0220-z
  • [13] Li, D., Qi, Y. and Rosalsky, A. (2012). On Jiang’s asymptotic distribution of the largest entry of a sample correlation matrix. J. Multivariate Anal. 111 256–270.
    Zentralblatt MATH: 1275.62037
    Digital Object Identifier: doi:10.1016/j.jmva.2012.04.002
  • [14] Li, D. and Rosalsky, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 423–447.
    Zentralblatt MATH: 1098.60034
    Digital Object Identifier: doi:10.1214/105051605000000773
    Project Euclid: euclid.aoap/1141654293
  • [15] Linnik, Y. V. (1961). On the probability of large deviations for the sums of independent variables. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 289–306. Univ. California Press, Berkeley, CA.
  • [16] Liu, W.-D., Lin, Z. and Shao, Q.-M. (2008). The asymptotic distribution and Berry–Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18 2337–2366.
    Zentralblatt MATH: 1154.60021
    Digital Object Identifier: doi:10.1214/08-AAP527
    Project Euclid: euclid.aoap/1227708921
  • [17] Muirhead, R. J. (2009). Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Statistics 197. Wiley, New York.
    Zentralblatt MATH: 0556.62028
  • [18] Shao, Q.-M. and Wang, Q. (2013). Self-normalized limit theorems: A survey. Probab. Surv. 10 69–93.
    Zentralblatt MATH: 1286.60029
    Digital Object Identifier: doi:10.1214/13-PS216
  • [19] Shao, Q.-M. and Zhou, W.-X. (2014). Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices. Ann. Probab. 42 623–648.
    Zentralblatt MATH: 1354.60020
    Digital Object Identifier: doi:10.1214/13-AOP837
    Project Euclid: euclid.aop/1393251298
  • [20] Zhou, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345–5363.
    Zentralblatt MATH: 1130.60032
    Digital Object Identifier: doi:10.1090/S0002-9947-07-04192-X

The Institute of Mathematical Statistics

The Annals of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more