Test of independence using generalized distance correlation
研究两个随机向量独立性检验的统计推断问题，提出一种适用于任意维度随机向量的广义距离协方差统一分布理论，并给出半置换程序估计渐近零分布。
We study the fundamental statistical inference concerning the testing of independence between two random vectors. Existing asymptotic theories for test statistics based on distance covariance can only apply to either low-dimensional or high-dimensional settings and require stringent distributional assumptions. In this work we develop a new unified distributional theory of the sample generalized distance covariance that works for random vectors of arbitrary dimensions under fairly mild moment conditions. In particular, a Gaussian approximation result is established with a nonasymptotic error bound, and the asymptotic null distribution of the sample generalized distance covariance is shown to be distributed as a linear combination of independently and identically distributed chi-squared random variables. To estimate the asymptotic null distribution practically, we propose a half-permutation procedure and provide the theoretical justification for its validity. The exact asymptotic distribution of the resampling distribution is derived under general marginal moment conditions, and the proposed procedure is shown to be asymptotically equivalent to the oracle procedure with known marginal distributions.