Estimation - Note 6

Estimation based on large average for models on trees

Stefka Asenova

2021-12-20

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In an unpublished note of Johan Segers (Segers (2019)) it is shown that if \(X= (X_1, \ldots, X_d)\) with unit Pareto margins and in the domain of attraction of a Huesler-Reiss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

The method of moments estimator is given by

\[ \hat{\theta}^{\mathrm{MMave}}_{k,n}=\arg\min_{\theta\in (0,\infty)^{|E|}} \|\hat{\Sigma}-\bar{\Sigma}(\theta)\|^2_F \]

\[\begin{equation*} \hat{\Sigma} = \frac{1}{|I|}\sum_{i\in I}(\Delta_{v,i}-\hat{\mu}, v\in U) (\Delta_{v,i}-\hat{\mu}, v\in U)^\top\, . %\end{split} \end{equation*}\]

A non-parametric estimator of this type \(\hat{\mu}\) and \(\hat{\Sigma}\) has been suggested in Engelke et al. (2015).

The composite likelihood estimator

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 \(\bar{\Sigma}\).

References

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.