# Estimation - Note 1" In gremes: Estimation of Tail Dependence in Graphical Models

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(gremes)  For application of this estimator, see Vignette "Code - Note 1". The MME is described in detail in Section 4.1 in @asenova2021. The idea is to find$(\theta_e, e\in E)$which minimizes the distance between the empirical and the theoretical covariance matrices: \hat{\theta}^{\mathrm{MM}}{n,k} = \arg\min{\theta\in(0,\infty)^{E}} \sum_{u\in U} \| \hat{\Sigma}{W_u, u}-\Sigma{W_u,u}(\theta) \|_F^2\, . where •$n$is the number of all observations in the sample •$k$is the number of the upper order statistics used in the estimation •$u$is the node for which we condition on the event${X_u>t}$•$\| \cdot \|_F$is the Frobenius norm •$U\subseteq V$is the set of observable variables •$W_u$is a subset on the node set depending on$u$. Typically a neighborhood of$u$or the nodes that are flow connected to$u$or the intersection of both. Note that the induced graph on$W_u$must be connected. A good practice is to compose the sets such that within each subset all parameters are uniquely identifiable. This means that every node in$W$with latent variable should be connected to at least three other nodes in the same set$W$. •$\hat{\Sigma}_{W_u, u}$is the non-parametric covariance matrix •$\Sigma_{W_u,u}(\theta)$is the parametric covariance matrix • For fixed$u$and$W_u$the parametric matrix$\Sigma_{W_u,u}$is given by \label{eq:hrdist} \big(\Sigma_{W,u}(\Lambda)\big){ij} = 2(\lambda{iu}^2 + \lambda_{ju}^2 - \lambda^2_{ij}), \qquad i,j\in W\setminus u. with \big(\Lambda(\theta)\big){ij} = \lambda^2{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , \qquad i,j\in V, \ i \ne j, e\in E. (See also he parameterization used for block graphs in Vignette "Introduction".) • If the sample of the original variables is$\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)$. \begin{equation} \hat{X}{v,i} = \frac{1}{1-\hat{F}{v,n}(\xi_{v,i})}, \qquad v \in U, \quad i = 1, \ldots, n. \end{equation} • Fix$u$and$W_u$. For given$k\in {1,\ldots n}$consider the set of indices [ I_{u} = {i = 1,\ldots,n: \hat{X}_{u,i} > n/k} ] • For every$v\in W_u\setminus u$and$i\in I_u$compose the differences \Delta_{uv,i} = \ln\hat{X}{v,i}-\ln\hat{X}{u,i}. • The vector of means of these differences is given by \begin{equation} \hat{\mu}{W_u,u} = \frac{1}{|I_u|}\sum{i\in I_u}(\Delta_{uv,i}, v\in W_u \setminus u). \end{equation} • The non-parametric covariance matrix$\hat{\Sigma}_{W_u,u}$is given by \begin{equation} \hat{\Sigma}{W_u,u} = \frac{1}{|I_u|}\sum{i\in I_u}(\Delta_{uv,i}-\hat{\mu}{W_u,u}, v\in W_u\setminus u) (\Delta{uv,i}-\hat{\mu}_{W_u,u}, v\in W_u\setminus u)^\top\, . %\end{split} \end{equation} An estimator of this type$\hat{\mu}$and$\hat{\Sigma}\$ has been suggested in @engelke.

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gremes documentation built on Feb. 16, 2023, 8:06 p.m.