Copula-based measures of reflection and permutation asymmetry and statistical tests

We propose measures of copula reflection and permutation asymmetry for data with positive quadrant dependence. We first define the measures of reflection asymmetry using a weighting function and then extend this approach to construct measures of permutation asymmetry for bivariate data. We define the corresponding statistical tests based on these measures and find that the proposed tests have higher statistical power comparing to some other tests for permutation and reflection symmetry studied in the literature. In addition, the measures can be used to summarize dependence structure of a multivariate data set in a few numbers and to select a more appropriate copula in the model.