Strength of tail dependence based on conditional tail expectation

Abstract

We use the conditional distribution and conditional expectation of one random variable given the other one being large to capture the strength of dependence in the tails of a bivariate random vector. We study the tail behavior of the boundary conditional cumulative distribution function (cdf) and two forms of conditional tail expectation (CTE) for various bivariate copula families. In general, for nonnegative dependence, there are three levels of strength of dependence in the tails according to the tail behavior of CTEs: asymptotically linear, sub-linear and constant. For each of these three levels, we investigate the tail behavior of CTEs for the marginal distributions belonging to maximum domain of attraction of Fréchet and Gumbel, respectively, and for copula families with different tail behavior.