## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 1006-1028.

### A consistent test of independence based on a sign covariance related to Kendall’s tau

#### Abstract

The most popular ways to test for independence of two ordinal random variables are by means of Kendall’s tau and Spearman’s rho. However, such tests are not consistent, only having power for alternatives with “monotonic” association. In this paper, we introduce a natural extension of Kendall’s tau, called $\tau^{*}$, which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, $\tau^{*}$ is the simplest, similarly to Kendall’s tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average $p$-values.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 1006-1028.

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393594014

Digital Object Identifier
doi:10.3150/13-BEJ514

Mathematical Reviews number (MathSciNet)
MR3178526

Zentralblatt MATH identifier
06291830

#### Citation

Bergsma, Wicher; Dassios, Angelos. A consistent test of independence based on a sign covariance related to Kendall’s tau. Bernoulli 20 (2014), no. 2, 1006--1028. doi:10.3150/13-BEJ514. https://projecteuclid.org/euclid.bj/1393594014

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#### Supplemental materials

• Supplementary material: A shorter proof of the main theorem for the continuous case and some miscellaneous further results. The supplement contains the following results: (i) a shorter proof of the main theorem, but only for the continuous case, (ii) the Cramér von Mises test as a special case, (iii) a shorter proof of main theorem for the case that one of the variables is binary, and (iv) a result for an extension to the case of variables in metric spaces.