Estimation - Note 3"
In gremes: Estimation of Tail Dependence in Graphical Models

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For application of this estimator, see Vignette "Code - Note 3".

Extremal coefficient estimator (ECE)

The pairwise extremal coefficients estimator is introduced in @eks16, is based on the bivariate stable tail dependence function (stdf). It is described in Section 4.3 in @asenova2021. More on the stdf can be found in @haan.

For the Huesler--Reiss distribution with parameter matrix $\Lambda(\theta)$ and for a pair of nodes $J = {u, v}$, the bivariate extremal coefficient is just \begin{equation} l_J(1, 1) = 2 \Phi(\lambda_{uv}(\theta)), \end{equation} with $\Phi$ the standard normal cumulative distribution function (cdf) and \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}

The non-parametric estimator of the stdf dates back to @drees1998 and yields the following estimator for the extremal coefficient $l_J(1, \ldots, 1)$ for $J \subseteq V$: \begin{equation} \label{eqn:lJkn} \hat{l}{J;n,k}(1,\ldots,1) = \frac{1}{k}\sum{i=1}^n \mathbb{1}\left( \max_{j \in J} n\hat{F}{j,n}(\xi{j,i}) >n+1/2-k \right). \end{equation}

Let $\mathcal{Q} \subseteq { J \subseteq U : |J| = 2 }$ be a collection of pairs of nodes associated to observable variables and put $q = |\mathcal{Q}|$, ensuring that $q \ge |E| = d-1$, the number of free edge parameters. The pairwise extremal coefficients estimator (ECE) of $\theta$ is \begin{equation} \label{eqn:ECE} \hat{\theta}^{\mathrm{ECE}}{n,k} = \arg\min{\theta \in (0,\infty)^{|E|}} \sum_{J \in \mathcal{Q}} \left( \hat{l}_{J;n,k}(1,1) - l_J(1, 1;\theta) \right)^2. \end{equation}

The $\mathcal{Q}$ must be the collection of all possible pairs of nodes in $U$. One may include also tri-variate extremal coefficients, in which case we will have $|J|=3$.

$n$ is the number of all observations in the sample
$k$ is the number of the upper order statistics used in the estimation
If the sample of the original variables is $\xi_{v,i}, v\in U, i=1,\ldots, n$ consider the transformation using the empirical cumulative distribution function $\hat{F}{v,n}(x)=\big[\sum{i=1}^n\mathbb{1}(\xi_{v,i}\leq x)\big]/(n+1)$.

ECE Version 2

The second version of the ECE which is implemented with object of class EKS_part uses the subsets $W_u$ for every $u\in U$ where $U$ is the set of noted with observable variables. It is similar to the MLE Version 1 explained in Vignette "Estimation - Note 2".

For fixed $u$ and $W_u$ such that $G(W_u)$ is a connected subgraph we apply the EC estimator of $\theta_{W_u}$ which is the collection of all edge weights within the subgraph $G(W_u)$.

In the first step we solve for every $u\in U$ and given $W_u$ \begin{equation} \hat{\theta}{W_u,n,k} = \arg\min{\theta \in (0,\infty)^{|W_u|-1}} \sum_{J \in \mathcal{Q_u}} \left( \hat{l}_{J;n,k}(1,1) - l_J(1, 1;\theta) \right)^2. \end{equation}
where $\mathcal{Q_u}$ is the collection of all possible pairs (and possibly triples) of nodes in $W_u$.

In a second step combine all estimates to obtain one estimate of $(\theta_e, e\in E)$

\begin{equation} \hat{\theta}^{\mathrm{ECEp}}{k,n} = \min{\theta\in [0,\infty)^{|E|}} \sum_{u\in U}\sum_{e\in E}(\hat{\theta}{e,W_u}-\theta{e})^2. \end{equation}