This estimator differs from the others because the conditioning event does not depend on a particular node \(u\) but it depends on the event that the geometric mean exceeds a high threshold.
For application of this estimator, see Vignette “Code - Note 6”.
In an unpublished note of Johan Segers (Segers (2019)) it is shown that if \(X= (X_1, \ldots, X_d)\) with unit Pareto margins and in the domain of attraction of a Huesler-Reiss distribution with parameter matrix \(\Lambda=(\lambda^2)_{ij}\), then it holds
\[\begin{equation}
\mathcal{L}\Big((Y_v-\bar{Y})_{v=1}^d|\bar{Y}>y\Big)
\rightarrow
\mathcal{N}_d(\bar{\mu}, \bar{\Sigma}),
\end{equation}\] with \(Y=(Y_1,\ldots, Y_d)=(\ln X_1,\ldots, \ln X_d )\) and \[\bar{\Sigma} =-M_d\Lambda M_d,\qquad
\bar{\mu}=-(1/d)\Lambda 1_d + (1/d)1_d^T \Lambda 1_d 1_d\] where \(M_d=I_d-(1/d)1_d1_d^T\), \(I_d\) is an identity matrix of size \(d\) and \(1_d\) is a a vector of ones of length \(d\).
Consider a tree \(T=(V,E)\) and edge weights \(\theta=(\theta_e, e\in E)\). Under the assumption that \(X=(X_v, v\in V)\) is in the domain of attraction of a Huesler-Reiss copula with unit Frechet margins and structured parameter matrix \(\Lambda(\theta)\) \[\begin{equation} \big(\Lambda(\theta)\big)_{ij} = \lambda^2_{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , \qquad i,j\in V, \ i \ne j, e\in E. \end{equation}\] we can employ the method of moments or the composite likelihood method to estimate \(\theta=(\theta_e, e\in E)\) from \(\bar{\Sigma}(\theta)\).
The method of moments estimator is given by
\[ \hat{\theta}^{\mathrm{MMave}}_{k,n}=\arg\min_{\theta\in (0,\infty)^{|E|}} \|\hat{\Sigma}-\bar{\Sigma}(\theta)\|^2_F \]
\(n\) is the number of all observations in the sample
\(k\) is the number of the upper order statistics used in the estimation
\(\| \cdot \|_F\) is the Frobenius norm
\(U\subseteq V\) is the set of observable variables
\(\hat{\Sigma}\) is the non-parametric covariance matrix
\(\bar{\Sigma}(\theta)\) is the parametric covariance matrix
The parametric matrix \(\bar{\Sigma}(\theta)\) is given by \[\begin{equation} \big(\bar{\Sigma}(\theta)\big)_{ij} = -M\big(\Lambda(\theta)\big)_{i,j\in U}M \end{equation}\] with \[\begin{equation} \big(\Lambda(\theta)\big)_{ij} = \lambda^2_{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , i \ne j, e\in E. \end{equation}\] (See also the parameterization used for trees 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 \(\hat{F}_{v,n}(x)=\big[\sum_{i=1}^n\mathbb{1}(\xi_{v,i}\leq x)\big]/(n+1)\). \[\begin{equation*} \hat{X}_{v,i} = \frac{1}{1-\hat{F}_{v,n}(\xi_{v,i})}, \qquad v \in U, \quad i = 1, \ldots, n. \end{equation*}\] Then consider their logarithm \[ \hat{Y}_{v,i}=\ln \hat{X}_{v,i} \]
For given \(k\in \{1,\ldots n\}\) consider the set of indices \[ I = \Big\{i = 1,\ldots,n: \overline{\hat{Y}}_{i}=(1/|U|)\sum_{v\in U} \hat{Y}_{v,i}> n/k \Big\} \]
For every \(v\in U\) and \(i\in I\) compose the differences \[\begin{equation} \Delta_{v,i} = \hat{Y}_{v,i}-\overline{\hat{Y}}_{i}. \end{equation}\]
The vector of means of these differences is given by \[\begin{equation*} \hat{\mu} = \frac{1}{|I|}\sum_{i\in I}(\Delta_{v,i}, v\in U ). \end{equation*}\]
The non-parametric covariance matrix \(\hat{\Sigma}\) is given by
\[\begin{equation*} \hat{\Sigma} = \frac{1}{|I|}\sum_{i\in I}(\Delta_{v,i}-\hat{\mu}, v\in U) (\Delta_{v,i}-\hat{\mu}, v\in U)^\top\, . %\end{split} \end{equation*}\]
A non-parametric estimator of this type \(\hat{\mu}\) and \(\hat{\Sigma}\) has been suggested in Engelke et al. (2015).
The composite likelihood estimator is given by \[ \hat{\theta}^{\mathrm{MLEave}}_{k,n}=\arg\max_{\theta\in(0,\infty)^{|E|}} L\Big(\bar{\mu}(\theta), \bar{\Sigma}(\theta); \{\Delta_{v,i}, i\in I, v\in U\}\Big). \] The likelihood function \(L\) above is the one of \(|U|\)-variate Gaussian probability density function with mean \(\bar{\mu}\) and covariance matrix \(\bar{\Sigma}\).
Engelke, S., A. Malinowski, Z. Kabluchko, and M. Schlather. 2015. “Estimation of Hüsler-Reiss Distributions and Brown-Resnick Processes.” Journal of the Royal Statistical Society. B 77: 239–65.
Segers, Johan. 2019. “On the Property of the Domain of Attraction of the Simple Huesler-Reiss Distribution: Lognormal Limit When Conditioning on the Geometric Mean Being Large.” Unpublished - Contact the Author: Johan.segers@uclouvain.be.