Copula-based measures of extreme dependence

We propose new measures of extreme dependence based on Kullback–Leibler relative entropy and defined in terms of the copula density. By reformulating tail exceedance functionals through the copula, we relocate the problem to the unit square and avoid the boundary and tail–instability issues inherent in kernel density methods traditionally used in the entropy literature. For estimation, we employ the Bernstein copula density estimator and show that, under standard mixing and smoothing conditions, its approximation bias is asymptotically negligible, allowing us to obtain consistent nonparametric estimators together with their Bahadur-type representations. These results lead to an asymptotically pivotal test for symmetric tail dependence whose limiting distribution is free of tuning parameters. We analyze the properties of the test under both global and local alternatives and establish the validity of a bootstrap version suitable for finite samples. A Monte Carlo simulation study shows that the bootstrap-based tests exhibit good finite-sample size and power properties across various data-generating processes and sample sizes. Finally, we present an empirical application that highlights the practical utility of the extreme dependence measures. Specifically, we quantify the degree of extreme dependence between the US financial market and several developed and emerging financial markets.