Tail Dependence of the Gaussian Copula Revisited

Paths and Indices of Maximal Tail Dependence

“…• searching for critical points of the function x → C(x, u 2 /x) over the interval [u 2 , 1], and for each u ∈ [0, 1], and then • checking which of the solutions is/are global maximum/maxima. Accomplishing these tasks may sometimes be relatively easy (Section 4), sometimes challenging (Furman et al, 2015), and in some cases obtaining closedform solutions may not even be possible (Section 6). Sometimes, especially when formulas for conditional copulas are readily available, it is useful to recall that partial derivatives of copulas are conditional copulas, and thus the task of determining the set of critical points becomes equivalent to finding all the solutions in x ∈ [u 2 , 1] to the equation…”

“…For example, Equation (2.7) has played a pivotal role when handling the Gaussian copula by Furman et al (2015). Functions of maximal dependence may or may not be continuous, and this is related to uniqueness of the path of maximal dependence.…”

A Maximal Tail Dependence-Based Clustering Procedure for Financial Time Series and Its Applications in Portfolio Selection

“…We refer to Furman et al (2015) for examples of expressions for λ * L and κ * L with closed forms in the case of parametric families of distributions. Notably, Furman et al (2016) proved that, in the Gaussian case, the classical and maximal tail dependence coefficients coincide. In the present paper, however, to speed up practical calculations, we resort to non-parametric approach in the following sections.…”

A Statistical Methodology for Assessing the Maximal Strength of Tail Dependence

“…As illustrated in the aforementioned paper, there can be several MTD paths but there is only one TOMD. We also know from the paper that the TOMD may or may not coincide with the TODD, and the Gaussian copula is one of those rare cases when the two tail orders are the same (Furman et al, 2016).…”