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TESTING FOR CHANGES IN KENDALL’S TAU

  • Herold Dehling (a1), Daniel Vogel (a2), Martin Wendler (a3) and Dominik Wied (a4)
Abstract

For a bivariate time series ((X i ,Y i )) i=1,...,n , we want to detect whether the correlation between X i and Y i stays constant for all i = 1,...n. We propose a nonparametric change-point test statistic based on Kendall’s tau. The asymptotic distribution under the null hypothesis of no change follows from a new U-statistic invariance principle for dependent processes. Assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall’s tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson’s moment correlation. Contrary to Pearson’s correlation coefficient, it shows no loss in efficiency at heavy-tailed distributions, and is therefore particularly suited for financial data, where heavy tails are common. We assume the data ((X i ,Y i )) i=1,...,n to be stationary and P-near epoch dependent on an absolutely regular process. The P-near epoch dependence condition constitutes a generalization of the usually considered L p -near epoch dependence allowing for arbitrarily heavy-tailed data. We investigate the test numerically, compare it to previous proposals, and illustrate its application with two real-life data examples.

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Corresponding author
*Address correspondence to Daniel Vogel, Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK; e-mail: daniel.vogel@abdn.ac.uk.
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The authors wish to thank their colleague Roland Fried for several very stimulating discussions that motivated this paper. Moreover, we are grateful for helpful comments from the editors and referees, which substantially improved a previous version of the paper. We are also indebted to Alexander Dürre, who did a thorough proofreading of the manuscript. The research was supported in part by the Collaborative Research Grant 823 Statistical modelling of nonlinear dynamic processes of the German Research Foundation.

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