On the uniqueness of distance covariance
On the uniqueness of distance covariance
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DOI：
10.1016/j.spl.2012.08.007
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发表时间：
2012-12-01
影响因子：
0.8
通讯作者：
Rizzo, Maria L.
中科院分区：
文献类型：
--
作者：
Szekely, Gabor J.;Rizzo, Maria L.
距离协方差和距离相关性是非负实数，表征任意维度上随机向量的独立性。在这项工作中，我们证明距离协方差是唯一的，首先将协方差定义为加权 L-2 范数，衡量两个随机向量的联合特征函数与其边缘特征函数的乘积之间的距离。这些加权L-2范数的刚性运动不变性和尺度等方差意味着距离协方差的权函数是唯一的。 (C) 2012 Elsevier B.V. 保留所有权利。
Distance covariance and distance correlation are non-negative real numbers that characterize the independence of random vectors in arbitrary dimensions. In this work we prove that distance covariance is unique, starting from a definition of a covariance as a weighted L-2 norm that measures the distance between the joint characteristic function of two random vectors and the product of their marginal characteristic functions. Rigid motion invariance and scale equivariance of these weighted L-2 norms imply that the weight function of distance covariance is unique. (C) 2012 Elsevier B.V. All rights reserved.