The Annals of Statistics

Tracy–Widom limit for Kendall’s tau

Zhigang Bao

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Abstract

In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy–Widom law for its largest eigenvalue. It is the first Tracy–Widom law for a nonparametric random matrix model, and also the first Tracy–Widom law for a high-dimensional U-statistic.

Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3504-3532.

Dates
Received: March 2018
Revised: August 2018
First available in Project Euclid: 31 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1572487401

Digital Object Identifier
doi:10.1214/18-AOS1786

Mathematical Reviews number (MathSciNet)
MR4025750

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 62G10: Hypothesis testing
Secondary: 62H10: Distribution of statistics 15B52: Random matrices 62H25: Factor analysis and principal components; correspondence analysis

Keywords
Tracy–Widom law largest eigenvalue nonparametric statistics U-statistics random matrices

Citation

Bao, Zhigang. Tracy–Widom limit for Kendall’s tau. Ann. Statist. 47 (2019), no. 6, 3504--3532. doi:10.1214/18-AOS1786. https://projecteuclid.org/euclid.aos/1572487401


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References

  • [1] Bai, Z. and Zhou, W. (2008). Large sample covariance matrices without independence structures in columns. Statist. Sinica 18 425–442.
  • [2] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • [3] Bandeira, A. S., Lodhia, A. and Rigollet, P. (2017). Marčenko–Pastur law for Kendall’s tau. Electron. Commun. Probab. 22 Paper no. 32.
  • [4] Bao, Z., Lin, L.-C., Pan, G. and Zhou, W. (2015). Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application. Ann. Statist. 43 2588–2623.
  • [5] Bao, Z., Pan, G. and Zhou, W. (2012). Tracy–Widom law for the extreme eigenvalues of sample correlation matrices. Electron. J. Probab. 17 Paper no. 88.
  • [6] Bao, Z., Pan, G. and Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. Ann. Statist. 43 382–421.
  • [7] Bao, Z. (2019). Supplement to “Tracy–Widom limit for Kendall’s tau.” DOI:10.1214/18-AOS1786SUPP.
  • [8] Bao, Z. G. (2018). Tracy–Widom limit for Spearman’s rho. Preprint.
  • [9] Bloemendal, A., Knowles, A., Yau, H.-T. and Yin, J. (2016). On the principal components of sample covariance matrices. Probab. Theory Related Fields 164 459–552.
  • [10] Ding, X. and Yang, F. (2018). A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices. Ann. Appl. Probab. 28 1679–1738.
  • [11] El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 2077–2117.
  • [12] El Karoui, N. (2007). Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35 663–714.
  • [13] Erdős, L., Knowles, A. and Yau, H.-T. (2013). Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincaré 14 1837–1926.
  • [14] Erdős, L., Yau, H.-T. and Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 341–407.
  • [15] Erdős, L., Yau, H.-T. and Yin, J. (2012). Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229 1435–1515.
  • [16] Fan, Z. and Johnstone, I. (2017). Tracy–Widom at each edge of real covariance estimators. Available at arXiv:1707.02352.
  • [17] Gao, J., Han, X., Pan, G. and Yang, Y. (2017). High dimensional correlation matrices: The central limit theorem and its applications. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 677–693.
  • [18] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19 293–325.
  • [19] Jiang, T. (2004). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880.
  • [20] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [21] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [22] Johnstone, I. M. and Ma, Z. (2012). Fast approach to the Tracy–Widom law at the edge of GOE and GUE. Ann. Appl. Probab. 22 1962–1988.
  • [23] Knowles, A. and Yin, J. (2017). Anisotropic local laws for random matrices. Probab. Theory Related Fields 169 257–352.
  • [24] Lee, J. O. and Schnelli, K. (2015). Edge universality for deformed Wigner matrices. Rev. Math. Phys. 27 1550018.
  • [25] Lee, J. O. and Schnelli, K. (2016). Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Ann. Appl. Probab. 26 3786–3839.
  • [26] Lee, J. O. and Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. Probab. Theory Related Fields 171 543–616.
  • [27] Ma, Z. (2012). Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices. Bernoulli 18 322–359.
  • [28] Marchenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. USSR, Sb. 1 457.
  • [29] Onatski, A. (2008). The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18 470–490.
  • [30] Pillai, N. S. and Yin, J. (2012). Edge universality of correlation matrices. Ann. Statist. 40 1737–1763.
  • [31] Pillai, N. S. and Yin, J. (2014). Universality of covariance matrices. Ann. Appl. Probab. 24 935–1001.
  • [32] Wang, K. (2012). Random covariance matrices: Universality of local statistics of eigenvalues up to the edge. Random Matrices Theory Appl. 1 1150005.
  • [33] Zhou, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345–5363.

Supplemental materials

  • Supplement to “Tracy–Widom limit for Kendall’s tau”. The supplement includes the proofs of some technical lemmas and some additional simulation studies.