Electronic Journal of Statistics

Large-sample theory for the Bergsma-Dassios sign covariance

Preetam Nandy, Luca Weihs, and Mathias Drton

Full-text: Open access


The Bergsma-Dassios sign covariance is a recently proposed extension of Kendall’s tau. In contrast to tau or also Spearman’s rho, the new sign covariance $\tau^{*}$ vanishes if and only if the two considered random variables are independent. Specifically, this result has been shown for continuous as well as discrete variables. We develop large-sample distribution theory for the empirical version of $\tau^{*}$. In particular, we use theory for degenerate U-statistics to derive asymptotic null distributions under independence and demonstrate in simulations that the limiting distributions give useful approximations.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2287-2311.

Received: February 2016
First available in Project Euclid: 29 August 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62G30: Order statistics; empirical distribution functions

Test of independence asymptotics U-statistics nonparametric correlation degeneracy Hoeffding’s D


Nandy, Preetam; Weihs, Luca; Drton, Mathias. Large-sample theory for the Bergsma-Dassios sign covariance. Electron. J. Statist. 10 (2016), no. 2, 2287--2311. doi:10.1214/16-EJS1166. https://projecteuclid.org/euclid.ejs/1472498028

Export citation


  • [1] W. Bergsma and A. Dassios. A consistent test of independence based on a sign covariance related to Kendall’s tau., Bernoulli, 20(2) :1006–1028, 2014.
  • [2] J. R. Blum, J. Kiefer, and M. Rosenblatt. Distribution free tests of independence based on the sample distribution function., Ann. Math. Statist., 32:485–498, 1961.
  • [3] S. S. Dhar, A. Dassios, and W. Bergsma. A study of the power and robustness of a new test for independence against contiguous alternatives., Electron. J. Statist., 10(1):330–351, 2016.
  • [4] E. B. Dynkin and A. Mandelbaum. Symmetric statistics, Poisson point processes, and multiple Wiener integrals., Ann. Statist., 11(3):739–745, 1983.
  • [5] Y. Heller and R. Heller. Computing the Bergsma Dassios sign-covariance. Available at:, arXiv:1605.08732, 2016.
  • [6] W. Hoeffding. A non-parametric test of independence., Ann. Math. Statist., 19:546–557, 1948.
  • [7] M. G. Kendall. A new measure of rank correlation., Biometrika, 30(1/2): pp. 81–93, 1938.
  • [8] R Core Team., R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2015. URL https://www.R-project.org/.
  • [9] M. L. Rizzo and G. J. Szekely., energy: E-statistics (energy statistics), 2014. URL http://CRAN.R-project.org/package=energy. R package version 1.6.2.
  • [10] R. J. Serfling., Approximation theorems of mathematical statistics. John Wiley & Sons, Inc., New York, 1980. Wiley Series in Probability and Mathematical Statistics.
  • [11] C. Spearman. The proof and measurement of association between two things., Am. J. Psychol., 15:72–101, 1904.
  • [12] G. J. Székely, M. L. Rizzo, and N. K. Bakirov. Measuring and testing dependence by correlation of distances., Ann. Statist., 35(6) :2769–2794, 2007.
  • [13] A. W. van der Vaart., Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998.
  • [14] L. Weihs., TauStar: Efficient Computation and Testing of the Bergsma-Dassios Sign Covariance, 2016. URL https://CRAN.R-project.org/package=TauStar. R package version 1.1.0.
  • [15] L. Weihs, M. Drton, and D. Leung. Efficient computation of the Bergsma–Dassios sign covariance., Comput. Statist., 31(1):315–328, 2016.