Huesler-Reiss distributions

Stefka Asenova

2023-02-11

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, 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.

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”.

References

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.