# Upper tail dependence of inverted max-stable distributions

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

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

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.

## Citation

BibTeX 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}
}