Large-Sample Theory for the Bergsma-Dassios Sign Covariance

By Preetam Nandy, Luca Weihs, Mathias Drton. 2016

35 citations recorded by the source corpus.

The Bergsma-Dassios sign covariance is a recently proposed extension of Kendall's tau. In contrast to tau or also Spearman's rho, the new sign covariance $\tau^*$ vanishes if and only if the two considered random variables are independent. Specifically, this result has been shown for continuous as well as discrete variables. We develop large-sample distribution theory for the empirical version of $\tau^*$. In particular, we use theory for degenerate U-statistics to derive asymptotic null distributions under independence and demonstrate in simulations that the limiting distributions give useful approximations.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Kernel‐based tests for joint independence (2016, methodology): “proaches for estimating or testing pairwise (in-)dependence including distance correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton, 2016; Nandy et al., 2016) or also non- and semiparametric copula-based correlations (Liu et al., 2012; Xue”
Testing independence in high dimensions with sums of rank correlations (2015, background): “He assumed higher-order moment conditions that were weakened in subsequent work [Zhou (2007), Liu, Lin and Shao (2008), Li, Liu and Rosalsky (2010), Li, Qi and Rosalsky (2012)].”
On the power of Chatterjee’s rank correlation (2020, background): “Indeed, applying Theorem 4.4 and Lemma 4.2 in Nandy et al. (2016) yields { ψv ( x ) , v = 1, 2, . . . } = { ψ(1)v1 ( x(1) ) ψ(2)v2 ( x(2) ) , v1, v2 = 1, 2, . . . } , where ψ(1)v1 ( x(1) ) ψ(2)v2 ( x(2) ) ≡ 2 cos { πv1FY (1) ( x(1) )} cos { πv2FY (2) ( x(2) )} is associated with eigenvalue λµv1,v2…”
A New Coefficient of Correlation (2019, background): “ls to address this deciency of the classical coecients [23], such as the maximal correlation coecient [5, 15, 21, 33], various coecients based on joint cumulative distribution functions and ranks [3, 4, 8, 9, 11, 14, 18, 22, 30, 32, 35, 36, 46{49], kernel-based methods [16, 17, 31, 40, 51], informat”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, unknown): “…H0, the eigenvalues λk, k ∈ Z+, are given by all the products λ1,j1λ2,j2 , j1, j2 ∈ Z+, where λ1,j, j ∈ Z+, and λ2,j, j ∈ Z+, are the nonzero eigenvalues of the integral equations E(dU(u, U)φ1(U)) = λ1φ1(u) and E(dV(v, V)φ2(V)) = λ2φ2(v), respectively (Nandy, Weihs, and Drton 2016, Lemma 4.2).”
High-dimensional consistent independence testing with maxima of rank correlations (2018, unknown): “= 3/(π 4i2j2), i, j ∈ Z+; see, e.g., Proposition 7 in Weihs et al. (2018) or Theorem 4.4 in Nandy et al. (2016).”