Upper tail dependence of inverted max-stable distributions

Definition 2 of Wadsworth and Tawn (2012) introduced inverted max-stable vectors as random vectors whose survival copula are max-stable. Max-stable vectors have upper tail dependence. The derivation below shows that inverted max-stable vectors are necessarily tail independent.

Proof. Suppose for simplicity that $\mathit{Z}=(Z_1,Z_2)$ is a simple max-stable bivariate random vector. The distribution function $G$ of a simple max-stable vector is $$\begin{array}{rl}G(z_1,z_2)& =\exp \{-V(z_1,z_2)\},\\ C(u_1,u_2)& =\exp [-V\{-\frac{1}{\log (u_1)},-\frac{1}{\log (u_2)}\}],\end{array}$$ where the exponent measure $V$ is an homogeneous function of order $-1$. Thus, transforming margins from unit Frechet to uniform and evaluating the copula at $u_1=u,u_2=u$, $$\begin{array}{rl}C(u,u)& =\exp [-V\{-\frac{1}{\log (u)},-\frac{1}{\log (u)}\}]\\ & =\exp \{-\log (u)\cdot V(1,1)\}\\ & =u^{\theta }.\end{array}$$ where $\theta \equiv V(1,1)$ is the extremal coefficient.

The associated inverted max-stable vector has copula $\tilde{C}(u,u)=C(1-u,1-u)$. We can therefore restrict attention to the study of the lower tail dependence index of $\mathit{Z}$,
$$\begin{array}{rl}{\lambda }_{\mathrm{L}}& =\underset{u\to 0}{lim}\frac{C(u,u)}{u}\\ & =\underset{u\to 0}{lim}{u}^{\theta -1}\\ & =\{\begin{array}{ll}0,& 1<\theta \le 2;\\ 1,& \theta =1.\end{array}\end{array}$$ The latter is possible if $C$ is the comonotone copula, in which case $Pr(U_1\le u,U_2\le u)=Pr(U<u)$. Since any bivariate max-stable vectors has upper tail dependence coefficient equal to $2-\theta$, the associated inverted max-stable vectors has lower tail dependence coefficient of $2-\theta$.

References

Wadsworth, Jennifer L., and Jonathan A. Tawn. 2012. “Dependence Modelling for Spatial Extremes.” Biometrika 99 (2): 253–72. https://doi.org/10.1093/biomet/asr080.