Nonparametric estimation of multivariate tail probabilities and tail dependence coefficients

Abstract

We propose three methods for estimating the joint tail probabilities based on a d-variate copula with dimension d≥2. For the first two methods, we use two different tail expansions of the copula which are valid under mild regularity conditions. We estimate the coefficients of these expansions using the maximum likelihood approach with appropriate data beyond a threshold in the tail. For the third method, we propose a family of tail-weighted measures of multivariate dependence and use these measures to estimate the coefficients of the second tail expansion using regression. This expansion is then used to estimate the joint tail probabilities when the empirical probabilities cannot be used because of lack of data in the tail. The three proposed methods can also be used to estimate tail dependence coefficients of a multivariate copula. Simulation studies are used to indicate when the methods give more accurate estimates of the tail probabilities and tail dependence coefficients. We apply the proposed methods to analyze tail properties of a data set of financial returns.