Estimating the coefficient of asymptotic tail independence

Many multivariate analyses require the account of extreme events. Correlation is an insufficient measure to quantify tail dependence. The most common tail dependence coefficients are based on the probability of simultaneous exceedances. The coefficient of asymptotic tail independence introduced in Ledford and Tawn (1996) is a bivariate measure often used in the tail modeling of data in finance, environment, insurance, among other fields of applications. It can be estimated as the tail index of the minimum component of a random pair with transformed unit Pareto marginals. The literature regarding the estimation of the tail index is extensive. Semi-parametric inference requires the choice of the number k of the largest order statistics that lead to the best estimate, where there is a tricky trade-off between variance and bias. Many methodologies have been developed to undertake this choice, most of them applied to the Hill estimator (Hill, 1975). We are going to analyze, through simulation, some of these methods within the estimation of the coefficient of asymptotic tail independence. We also compare with a minimum-variance reduced-bias Hill estimator presented in Caeiro et al. (2005). A pure heuristic procedure adapted from Frahm et al. (2005), used in a different context but with a resembling framework, will also be implemented. We will see that some of these simple tools should not be discarded in this context. Our study will be complemented by applications to real datasets.