Estimation - Note 1"

In gremes: Estimation of Tail Dependence in Graphical Models

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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: \begin{equation} \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\, . \end{equation}

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 \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} with \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} (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 \begin{equation} \Delta_{uv,i} = \ln\hat{X}{v,i}-\ln\hat{X}{u,i}. \end{equation}

• 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.

References

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