Huesler-Reiss distributions"
In gremes: Estimation of Tail Dependence in Graphical Models

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Parameterization on trees

The parameterization used for models on trees is the following \begin{equation} H_{\Lambda}(z) = \exp\left{- \sum_{u\in V} \frac{1}{z_u}\Phi_{|V|-1}\left( \ln\frac{z_v}{z_u} +2\lambda^2_{uv}, v\in V\setminus u; \Sigma_{V,u}(\Lambda) \right) \right}, \qquad z \in (0, \infty)^{|V|}, \end{equation} where $\Phi_p(\,\cdot\,; \Sigma)$ denotes the $p$-variate zero mean Gaussian cdf with covariance matrix $\Sigma$. This is a Huesler-Reiss copula with univariate Frechet margins. This expression is due to @nikoloulopoulos_2009, @genton2011 and @huser_dav2013. The matrix $\lambda_{ij}$ depends on $(\theta_e, e\in E)$, namely \begin{equation} \big(\Lambda(\theta)\big){ij} = \lambda^2{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , \qquad i,j\in V, \ i \ne j, e\in E. \end{equation} $p(i,j)$ is the unique path between nodes $i,j$. The matrix $\Sigma_{W,u}$ is given by \begin{equation} \label{eq:hrdist} \big(\Sigma_{W,u}(\Lambda)\big){ij} = 2(\lambda{iu}^2 + \lambda_{ju}^2 - \lambda^2_{ij}), \qquad i,j\in W\setminus u. \end{equation}

The bivariate Huesler-Reiss copula with Unit Frechet margins when the variables are adjacent and the edge weight between them is $\theta_e$ is given by
\begin{equation} %\begin{split} H_{\theta_e}(z_u, z_v) %\& = \exp\left{- \frac{1}{z_u}\Phi\left( \frac{\theta_e}{2}+\frac{\ln z_v/z_u}{\theta_e}\right) - \frac{1}{z_v}\Phi\left( \frac{\theta_e}{2}+\frac{\ln z_u/z_v}{\theta_e}\right) \right}, \qquad z_u, z_v \in (0, \infty)^2, %\end{split} \end{equation}

Such a parameterization means that large values of $\theta$'s or $\lambda$'s correspond to weak extremal dependence and small values to stronger extremal dependence.

The method estimate applied to objects of classes MME, MLE, MLE1, MLE2, EKS, EKS_part, EngHitz, MMEave, MLEave estimates $(\theta_e, e\in E)$. See also Vignettes "Code - Note" 1-4 and 6.

Parameterization on block graphs

The parameterization of the Huesler-Reiss distribution for models on block graphs is the following \begin{equation} %\begin{split} H_{\Lambda}(z) %\& = \exp\left{- \sum_{u\in V} \frac{1}{z_u}\Phi_{|V|-1}\left( \ln\frac{z_v}{z_u} +2\lambda^2_{uv}, v\in V\setminus u; \Sigma_{V,u}(\Lambda) \right) \right}, \qquad z \in (0, \infty)^{|V|}, %\end{split} \end{equation}

where the parameter $\lambda_{ij}^2, i,j \in V$ is defined in terms of the edge weights $\delta^2_{e}, e\in E$. The relation is given by \begin{equation} \big(\Lambda(\theta)\big){ij}=\lambda{ij}^2(\delta) = \sum_{e\in p(i,j)}\delta^2_{e} \end{equation} for $\delta=(\delta_e^2, e\in E)$ and $p(i,j)$ the unique shortest path between nodes $i,j$. The matrix $\Sigma_{W,u}$ is given by \begin{equation} \big(\Sigma_{W,u}(\Lambda)\big){ij} = 2(\lambda{iu}^2 + \lambda_{ju}^2 - \lambda^2_{ij}), \qquad i,j\in W\setminus u. \end{equation}

The bivariate Huesler-Reiss copula with Unit Frechet margins when the variables are adjacent and the edge weight between them is $\delta_e$ is given by
\begin{equation} %\begin{split} H_{\delta_e}(z_u, z_v) %\& = \exp\left{- \frac{1}{z_u}\Phi\left( \frac{\ln z_v/z_u}{2\delta_e}+\delta_e\right) - \frac{1}{z_v}\Phi\left( \frac{\ln z_u/z_v}{2\delta_e}+\delta_e\right) \right}, \qquad z_u, z_v \in (0, \infty)^2, %\end{split} \end{equation}

Such a parameterization means that large values of $\delta$'s or $\lambda$'s correspond to weak extremal dependence and small values to stronger extremal dependence.

The method estimate applied to objects of classes HRMBG estimates $(\delta^2_e, e\in E)$. See also Vignette "Code - Note 5".