Electronic Communications in Probability

Marčenko-Pastur law for Kendall’s tau

Afonso S. Bandeira, Asad Lodhia, and Philippe Rigollet

Full-text: Open access

Abstract

We prove that Kendall’s Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors $X_1, \ldots , X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.

Article information

Source
Electron. Commun. Probab. Volume 22 (2017), paper no. 32, 7 pp.

Dates
Received: 25 January 2017
Accepted: 15 May 2017
First available in Project Euclid: 3 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1496455233

Digital Object Identifier
doi:10.1214/17-ECP59

Zentralblatt MATH identifier
1366.60007

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 62H20: Measures of association (correlation, canonical correlation, etc.)

Keywords
statistics random matrix theory

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bandeira, Afonso S.; Lodhia, Asad; Rigollet, Philippe. Marčenko-Pastur law for Kendall’s tau. Electron. Commun. Probab. 22 (2017), paper no. 32, 7 pp. doi:10.1214/17-ECP59. https://projecteuclid.org/euclid.ecp/1496455233


Export citation

References

  • [1] Zhidong Bai and Jack W. Silverstein,Spectral analysis of large dimensional random matrices, second ed., Springer Series in Statistics, Springer, New York, 2010.
  • [2] Zhidong Bai and Wang Zhou,Large sample covariance matrices without independence structures in columns, Statist. Sinica18(2008), no. 2, 425–442.
  • [3] Francois Crénin, David Cressey, Sophie Lavaud, Jiali Xu, and Pierre Clauss,Random matrix theory applied to correlations in operational risk, Journal of Operational Risk10(2015), no. 4, 45–71.
  • [4] Víctor H. de la Peña and Evarist Giné,Decoupling, Probability and its Applications (New York), Springer-Verlag, New York, 1999.
  • [5] Fredrick Esscher,On a method of determining correlation from the ranks of the variates, Scandinavian Actuarial Journal1924(1924), no. 1, 201–219.
  • [6] Wassily Hoeffding,A class of statistics with asymptotically normal distribution, Ann. Math. Statistics19(1948), 293–325.
  • [7] M. G. Kendall,A new measure of rank correlation, Biometrika30(1938), no. 1/2, 81–93.
  • [8] Maurice Kendall and Jean Dickinson Gibbons,Rank correlation methods, fifth ed., A Charles Griffin Title, Edward Arnold, London, 1990.
  • [9] Jarl Waldemar Lindeberg,Über die korrelation, Den VI skandinaviske Matematikerkongres i København (1925), 437–446.
  • [10] JW Lindeberg,Some remarks on the mean error of the percentage of correlation, Nordic Statistical Journal1(1929), 137–141.
  • [11] Han Liu, Fang Han, Ming Yuan, John Lafferty, and Larry Wasserman,High-dimensional semiparametric Gaussian copula graphical models, Ann. Statist.40(2012), no. 4, 2293–2326.
  • [12] Sean O’Rourke,A note on the Marchenko-Pastur law for a class of random matrices with dependent entries, Electron. Commun. Probab.17(2012), no. 28, 13.
  • [13] Debashis Paul and Alexander Aue,Random matrix theory in statistics: a review, J. Statist. Plann. Inference150(2014), 1–29.
  • [14] Y. Q. Yin and P. R. Krishnaiah,Limit theorems for the eigenvalues of product of large-dimensional random matrices when the underlying distribution is isotropic, Teor. Veroyatnost. i Primenen.31(1986), no. 2, 394–398.