Abstract
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.
Highlights
Since the early part of the last century, several measures of association have been proposed to detect the association between random variables
Since the exact distributions of τn, τn∗ and dcovn are not tractable, we estimate the size and the power of the tests based on the asymptotic distributions of τn, τn∗ and dcovn. Since both the tests based on τn∗ and dcovn are consistent tests against fixed alternative, here we investigate the asymptotic powers of the tests under contiguous alternatives
As we have mentioned in the Introduction, the nonrobustness of distance covariance is expected to be reflected in the asymptotic power study, which will be fully discussed in the forthcoming section
Summary
Since the early part of the last century, several measures of association have been proposed to detect the association between random variables. In view of the fact that τn∗ → τ ∗ in probability as n → ∞, the bias of τn∗ will be bounded by 1/4 in probability when the data is obtained from the joint distribution function having independent marginal distribution functions. The assertion in Proposition 1 implies that τ is a robust measure in the sense of having bounded b(β; τ (F0)), whereas unlike b(β; τ (F0)) and b(β; τ ∗(F0)), b(β; dcov(F0)) is unbounded. This fact implies that distance covariance is nonrobust against the outliers. As we have mentioned in the Introduction, the nonrobustness of distance covariance is expected to be reflected in the asymptotic power study, which will be fully discussed in the forthcoming section
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