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Register a new userOn the power of Chatterjee rank correlation
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Added by
Fang Han
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much recent attention. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjees new correlation coefficient to three established rank correlations that also facilitate consistent tests of independence, namely, Hoeffdings $D$, Blum-Kiefer-Rosenblatts $R$, and Bergsma-Dassios-Yanagimotos $tau^*$. We contrast their computational efficiency in light of recent advances, and investigate their power against local rotation and mixture alternatives. Our main results show that Chatterjees coefficient is unfortunately rate sub-optimal compared to $D$, $R$, and $tau^*$. The situation is more subtle for a related earlier estimator of Dette et al. (2013). These results favor $D$, $R$, and $tau^*$ over Chatterjees new correlation coefficient for the purpose of testing independence.
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Chatterjee (2021)s ingenious approach to estimating a measure of dependence first proposed by Dette et al. (2013) based on simple rank statistics has quickly caught attention. This measure of dependence has the unusual property of being between 0 and 1, and being 0 or 1 if and only if the corresponding pair of random variables is independent or one is a measurable function of the other almost surely. However, more recent studies (Cao and Bickel, 2020; Shi et al., 2021b) showed that independence tests based on Chatterjees rank correlation are unfortunately rate-inefficient against various local alternatives and they call for variants. We answer this call by proposing revised Chatterjees rank correlations that still consistently estimate the same dependence measure but provably achieve near-parametric efficiency in testing against Gaussian rotation alternatives. This is possible via incorporating many right nearest neighbors in constructing the correlation coefficients. We thus overcome the only one disadvantage of Chatterjees rank correlation (Chatterjee, 2021, Section 7).
In this paper we propose a class of weighted rank correlation coefficients extending the Spearmans rho. The proposed class constructed by giving suitable weights to the distance between two sets of ranks to place more emphasis on items having low rankings than those have high rankings or vice versa. The asymptotic distribution of the proposed measures and properties of the parameters estimated by them are studied through the associated copula. A simulation study is performed to compare the performance of the proposed statistics for testing independence using asymptotic relative efficiency calculations.
Consider a Gaussian vector $mathbf{z}=(mathbf{x},mathbf{y})$, consisting of two sub-vectors $mathbf{x}$ and $mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by $Sigma_{mathbf{u}mathbf{v}}$ the population cross-covariance matrix of random vectors $mathbf{u}$ and $mathbf{v}$, and denote by $S_{mathbf{u}mathbf{v}}$ the sample counterpart. The canonical correlation coefficients between $mathbf{x}$ and $mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $Sigma_{mathbf{x}mathbf{x}}^{-1}Sigma_{mathbf{x}mathbf{y}}Sigma_{mathbf{y}mathbf{y}}^{-1}Sigma_{mathbf{y}mathbf{x}}$. In this paper, we focus on the case that $Sigma_{mathbf{x}mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1geqcdotsgeq r_k>0$. We study the sample counterparts of $r_i,i=1,ldots,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix $S_{mathbf{x}mathbf{x}}^{-1}S_{mathbf{x}mathbf{y}}S_{mathbf{y}mathbf{y}}^{-1}S_{mathbf{y}mathbf{x}}$, denoted by $lambda_1geqcdotsgeq lambda_k$. We show that there exists a threshold $r_cin(0,1)$, such that for each $iin{1,ldots,k}$, when $r_ileq r_c$, $lambda_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_i>r_c$, $lambda_i$ possesses an almost sure limit in $(d_{+},1]$. We also obtain the limiting distribution of $lambda_i$s under appropriate normalization. Specifically, $lambda_i$ possesses Gaussian type fluctuation if $r_i>r_c$, and follows Tracy-Widom distribution if $r_i<r_c$. Some applications of our results are also discussed.
We provide sufficient conditions for the asymptotic normality of the generalized correlation coefficient $sum a_{ij}b_{ij}$ under the permutation null distribution when $a_{ij}$s are symmetric and $b_{ij}$s are symmetric.
Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct distribution-free yet consistent tests of independence between random vectors. This is the problem addressed in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular examples. In a unified study, we derive a general asymptotic representation of center-outward rank-based test statistics under independence, extending to the multivariate setting the classical H{a}jek asymptotic representation results. This representation permits direct calculation of limiting null distributions and facilitates a local power analysis that provides strong support for the center-outward approach by establishing, for the first time, the nontrivial power of center-outward rank-based tests over root-$n$ neighborhoods within the class of quadratic mean differentiable alternatives.
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