Copulas, diagonals, and tail dependence

We present some known and novel aspects about bivariate copulas with prescribed diagonal section by highlighting their use in the description of the tail dependence. Moreover, we present the tail concentration function (which depends on the diagonal section of a copula) as a tool to give a description of tail dependence at finite scale. The tail concentration function is hence used to introduce a graphical tool that can help to distinguish different families of copulas in the copula test space. Moreover, it serves as a basis to determine the grouping structure of different financial time series by taking into account their pairwise tail behavior.