Copula-based models for the power of independence tests

We consider independence tests and the methods to evaluate their efficiency. First, we observe that many of the most used independence tests are functions of the empirical copula, which is a sufficient statistic. Hence, the power of these tests, such as the tests based oil Spearman's p, on Kendall's tau, and on Gini's y, depend solely on the theoretical copula, and not oil the marginal distributions. Then, we consider monotone dependence tests and we propose a parametric model to define the power function. Such a model is based oil a path of copulas, from the copula of discordance to the copula of concordance, and call be characterized by the copula of the underlying joint distribution. Moreover, we introduce a consistent estimator of the path of copulas. Finally, we provide some examples of applications, and in particular, a bootstrap-plug-in estimator of the power curve, all useful for power comparison.