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 \(2-\theta\).

## References

*Biometrika*99 (2): 253–72. https://doi.org/10.1093/biomet/asr080.

## Citation

```
@online{belzile2019,
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}
}
```