The distribution of Kendall's tau for testing the significance of cross-correlation in persistent data

Abstract Kendall's tau (τ) has been widely used as a distribution-free measure of cross-correlation between two variables. It has been previously shown that persistence in the two involved variables results in the inflation of the variance of τ. In this paper, the full null distribution of Kendall's τ for persistent data with multivariate Gaussian dependence is derived, and an approximation to the full distribution is proposed. The effect of the deviation from the multivariate Gaussian dependence model on the distribution of τ is also investigated. As a demonstration, the temporal consistency and field significance of the cross-correlation between the North Hemisphere (NH) temperature time series in the period 1850–1995 and a set of 784 NH tree-ring width (TRW) proxies in addition to 105 NH tree-ring maximum latewood density (MXD) proxies are studied. When persistence is ignored, the original Mann-Kendall test gives temporally inconsistent results between the early half (1850–1922) and the late half (1923–1995) of the record. These temporal inconsistencies are largely eliminated when persistence is accounted for, indicating the spuriousness of a large portion of the identified cross-correlations. Furthermore, the use of the modified test in combination with a field significance test that is robust to spatial correlation indicates the absence of field significant cross-correlation in both halves of the record. These results have serious implications for the use of tree-ring data as temperature proxies, and emphasize the importance of utilizing the correct distribution of Kendall's τ in order to avoid the overestimation of the significance of cross-correlation between data that exhibit significant persistence. Citation Hamed, K. H. (2011) The distribution of Kendall's tau for testing the significance of cross-correlation in persistent data. Hydrol. Sci. J. 56(5), 841–853.

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