Estimation - Note 1

Method of Moments Estimator for models on trees

Stefka Asenova

2021-12-20


For application of this estimator, see Vignette “Code - Note 1”.


The MME is described in detail in Section 4.1 in Asenova, Mazo, and Segers (2021). 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

\[\begin{equation*} \hat{X}_{v,i} = \frac{1}{1-\hat{F}_{v,n}(\xi_{v,i})}, \qquad v \in U, \quad i = 1, \ldots, n. \end{equation*}\]

\[\begin{equation*} \hat{\Sigma}_{W_u,u} = \frac{1}{|I_u|}\sum_{i\in I_u}(\Delta_{uv,i}-\hat{\mu}_{W_u,u}, v\in W_u\setminus u) (\Delta_{uv,i}-\hat{\mu}_{W_u,u}, v\in W_u\setminus u)^\top\, . %\end{split} \end{equation*}\]

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

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.

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.