## Distance multivariance: New dependence measures for random vectors. (arXiv:1711.07775v2 [math.PR] UPDATED)

We introduce two new measures for the dependence of \$n \ge 2\$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted \$L^2\$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Sz\'ekely, Rizzo and Bakirov) from pairs of random variables to \$n\$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of \$n\$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We introduce two new measures for the dependence of \$n \ge 2\$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted \$L^2\$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Sz\'ekely, Rizzo and Bakirov) from pairs of random variables to \$n\$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of \$n\$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.
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