Tail Copula Estimation for Heteroscedastic Extremes

Consider independent multivariate random vectors which follow the same copula, but where each marginal distribution is allowed to be non-stationary. This non-stationarity is for each marginal governed by a scedasis function (see Einmahl et al. (2016)) that is the same for all marginals. We establish the asymptotic normality of the usual rank-based estimator of the stable tail dependence function, or, when specialized to bivariate random vectors, the corresponding estimator of the tail copula. Remarkably, the heteroscedastic marginals do not affect the limiting process. Next, under a bivariate setup, we develop nonparametric tests for testing whether the scedasis functions are the same for both marginals. Detailed simulations show the good performance of the estimator for the tail dependence coefficient as well as that of the new tests. In particular, novel asymptotic confidence intervals for the tail dependence coefficient are presented and their good finite-sample behavior is shown. Finally an application to the S&P500 and Dow Jones indices reveals that their scedasis functions are about equal and that they exhibit strong tail dependence.