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, Joe, and Li (2009), Genton, Ma, and Sang (2011) and Huser and Davison (2013). 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.
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”.
Genton, M. G., Y. Ma, and H. Sang. 2011. “On the Likelihood Function of a Gaussian Max-Stable Processes.” Biometrika 98 (2): 481–88.
Huser, R., and A. C. Davison. 2013. “Composite Likelihood Estimation for the Brown–Resnick Process.” Biometrika 100 (2): 511–18.
Nikoloulopoulos, Aristidis K., Harra Joe, and Haijun Li. 2009. “Extreme Value Properties of Multivariate T Copulas.” Extremes 12: 129–48.