Estimation - Note 3

Extremal Coefficients Estimator for models on trees

Stefka Asenova

2021-12-20


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


Extremal coefficient estimator (ECE)

The pairwise extremal coefficients estimator is introduced in Einmahl, Kiriliouk, and Segers (2017), is based on the bivariate stable tail dependence function (stdf). It is described in Section 4.3 in Asenova, Mazo, and Segers (2021). More on the stdf can be found in Haan and Ferreira (2006).

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 Drees and Huang (1998) 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 \(|J|=3\).

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 \((\theta_e, e\in E)\)

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

References

Asenova, Stefka, Gildas Mazo, and Johan Segers. 2021. “Inference on Extremal Dependence in the Domain of Attraction of a Structured Hüsler–Reiss Distribution Motivated by a Markov Tree with Latent Variables.” Extremes. https://doi.org/10.1007/s10687-021-00407-5.

Drees, Holger, and Xin Huang. 1998. “Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function.” Journal of Multivariate Analysis 64 (1): 25–46. https://doi.org/https://doi.org/10.1006/jmva.1997.1708.

Einmahl, J., A. Kiriliouk, and J. Segers. 2017. “A Continuous Updating Weighted Least Squares Estimator of Tail Dependence in High Dimensions.” Extremes.

Haan, L. de, and A. Ferreira. 2006. Extreme Value Theory: An Introduction. Springer-Verlag New York.