Upper tail dependence of inverted max-stable distributions

This short note shows that inverted max-stable vectors cannot have upper tail dependence.

Tuesday, July 9, 2019

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 \(\boldsymbol{Z}= (Z_1, Z_2)\) is a simple max-stable bivariate random vector. The distribution function \(G\) of a simple max-stable vector is \[\begin{align*} G(z_1, z_2)&=\exp\left\{-V(z_1, z_2)\right\}, \\C(u_1, u_2)&=\exp\left[-V \left\{-\frac{1}{\log(u_1)}, -\frac{1}{\log(u_2)}\right\}\right], \end{align*}\] 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 \(u_1=u, u_2=u\), \[\begin{align*} C(u,u)&=\exp\left[-V \left\{-\frac{1}{\log(u)}, -\frac{1}{\log(u)}\right\}\right]\\&= \exp\left\{-\log(u) \cdot V(1,1)\right\}\\&=u^\theta. \end{align*}\] 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 \(\boldsymbol{Z}\),
\[\begin{align*} \lambda_{\mathrm{L}}&=\lim_{u \to 0} \frac{C(u,u)}{u}\\&=\lim_{u \to 0} u^{\theta-1}\\&= \begin{cases}0, & 1 < \theta \le 2; \\1, & \theta=1.\end{cases} \end{align*}\] The latter is possible if \(C\) is the comonotone copula, in which case \(\Pr({U_1 \leq u, U_2 \leq u})=\Pr({U <u})\). 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 \(2-\theta\).


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.


BibTeX citation:
  author = {Belzile, Léo},
  title = {Upper Tail Dependence of Inverted Max-Stable Distributions},
  date = {2019-07-09},
  url = {https://lbelzile.bitbucket.io/posts/taildep-inverted-max-stable},
  langid = {en}
For attribution, please cite this work as:
Belzile, Léo. 2019. “Upper Tail Dependence of Inverted Max-Stable Distributions.” July 9, 2019. https://lbelzile.bitbucket.io/posts/taildep-inverted-max-stable.