## New measure of the bivariate asymmetry.(English)Zbl 1465.62108

Summary: A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank-based distance correlation computed over two complementary order-preserved sets. General properties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth-order degenerate $$V$$-statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.

### MSC:

 62H20 Measures of association (correlation, canonical correlation, etc.) 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62G30 Order statistics; empirical distribution functions 60E10 Characteristic functions; other transforms

### Software:

IndependenceTests; TwoCop; QRM
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