Electronic Journal of Statistics

A study of the power and robustness of a new test for independence against contiguous alternatives

Subhra Sankar Dhar, Angelos Dassios, and Wicher Bergsma

Full-text: Open access


Various association measures have been proposed in the literature that equal zero when the associated random variables are independent. However many measures, (e.g., Kendall’s tau), may equal zero even in the presence of an association between the random variables. In order to overcome this drawback, Bergsma and Dassios (2014) proposed a modification of Kendall’s tau, (denoted as $\tau^{*}$), which is non-negative and zero if and only if independence holds. In this article, we investigate the robustness properties and the asymptotic distributions of $\tau^{*}$ and some other well-known measures of association under null and contiguous alternatives. Based on these asymptotic distributions under contiguous alternatives, we study the asymptotic power of the test based on $\tau^{*}$ under contiguous alternatives and compare its performance with the performance of other well-known tests available in the literature.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 330-351.

Received: June 2015
First available in Project Euclid: 17 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G35: Robustness 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Contiguous alternatives Distance covariance Kendall’s tau Pitman efficacy Robustness properties Test for independence


Dhar, Subhra Sankar; Dassios, Angelos; Bergsma, Wicher. A study of the power and robustness of a new test for independence against contiguous alternatives. Electron. J. Statist. 10 (2016), no. 1, 330--351. doi:10.1214/16-EJS1107. https://projecteuclid.org/euclid.ejs/1455715965

Export citation


  • Babu, G. J., & Rao, C. R. (1988). Joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population., Journal of Multivariate Analysis, 27(1), 15–23.
  • Banerjee, M. (2005). Likelihood ratio tests under local alternatives in regular semiparametric models., Statistica Sinica, 15(3), 635–644.
  • Behnen, K. (1971). Asymptotic optimality and ARE of certain rank-order tests under contiguity., The Annals of Mathematical Statistics, 325–329.
  • Behnen, K., & Neuhaus, G. (1975). A central limit theorem under contiguous alternatives., The Annals of Statistics, 3(6), 1349–1353.
  • Bergsma, W., & Dassios, A. (2014). A consistent test of independence based on a sign covariance related to Kendall’s tau., Bernoulli, 20(2), 1006–1028.
  • Bergsma, W. P. (2006). A new correlation coefficient, its orthogonal decomposition, and associated tests of independence., arXiv:math/0604627v1 [math.ST].
  • Blum, J. R., Kiefer, J., & Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function., The Annals of Mathematical Statistics, 32, 485–498.
  • Christensen, D. (2005). Fast algorithms for the calculation of Kendall’s $\tau$., Computational Statistics, 20(1), 51–62.
  • Gibbons, J. D., & Chakraborti, S. (2011)., Nonparametric statistical inference. Springer.
  • Gregory, G. G. (1977). Large sample theory for $U$-statistics and tests of fit., The Annals of Statistics, 110–123.
  • Hajek, J., Sidak, Z., & Sen, P. K. (1999)., Theory of rank tests. Academic Press.
  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution., The Annals of Mathematical Statistics, 19, 293–325.
  • Huber, P. J. (2011)., Robust statistics (2nd edition). Springer.
  • Kendall, M. G. (1938). A new measure of rank correlation., Biometrika, 30(1/2), 81–93.
  • Lee, A. (1990). $U$-statistics., Theory and Practice, Marcel Dekker, New York.
  • Rényi, A. (1959). On measures of dependence., Acta Math. Acad. Sci. Hung., 10, 441–451.
  • Serfling, R. J. (1980)., Approximation theorems of mathematical statistics. New York: Wiley-Blackwell.
  • Spearman, C. (1904). The proof and measurement of association between two things., The American Journal of Psychology, 15(1), 72–101.
  • Székely, G. J., Rizzo, M. L., & Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances., The Annals of Statistics, 35(6), 2769–2794.
  • Van der Vaart, A. W. (2000)., Asymptotic statistics (Vol. 3). Cambridge University Press.
  • Weihs, L., Drton, M., & Leung, D. (2016 (to appear)). Efficient computation of the Bergsma-Dassios sign covariance., Computational Statistics.