gremes - Introduction

Trees

Let \(T=(V,E)\) be a tree with node set \(V\) and edge set \(E\). Consider a \(|V|\)-variate random vector \(X=(X_v, v\in V)\) for which it holds:

• \(X\) satisfies the global Markov property with respect to the tree \(T\)

• every bivariate distribution between two adjacent variables is a bivariate Huesler-Reiss distribution with parameter \(\theta_e\) for some \(e\in E\). For the particular parameterization please refer to Vignette “Distributions”.

• standardized to unit-Pareto marginal distributions.

Based on different asymptotic results on \(X\) we can have different estimators of the parameters \(\theta_e, e\in E\) which we collect in a vector \(\theta\in (0,\infty)^{|E|}\).

1. Consider the limiting distribution of \[ \ln (X/X_u)\mid X_u>t, \qquad t\rightarrow\infty. \] It can be shown that the limiting distribution is multivariate Gaussian distribution with mean vector \(\mu_u(\theta)\) and covariance matrix \(\Sigma_{u}(\theta)\). One can consult Asenova, Mazo, and Segers (2021) and references therein. Based on this asymptotic result we come up with method of moments type estimator and a maximum likelihood (rather composite likelihood function) estimator which both aim at estimating \(\Sigma_{u}\) and accordingly \(\theta\).

2. Consider the limiting distribution of the componentwise maxima if we dispose of a random sample of size \(n\) of \(X\) \[ \Big(\frac{1}{n}\max_{i=1, \ldots, n}X_{v,i}, v\in V\Big), \qquad n\rightarrow \infty. \] The limit is a max-stable Huesler-Reiss copula with unit Frechet margins and with parameter matrix \(\Lambda\) given by \[\begin{equation} \big(\Lambda(\theta)\big)_{ij} = \lambda^2_{ij}(\theta) = \frac{1}{4}\sum_{e \in p(i,j)} \theta_e^2\, , \qquad i,j\in V, \ i \ne j, e\in E. \end{equation}\] he notation \(p(i,j)\) means the unique shortest path between \(i,j\). On the basis of this result we come up with an estimator using the stable tail dependence function associated to the max-stable Huesler-Reiss distribution. This estimator is introduced in Einmahl, Kiriliouk, and Segers (2017) and it is named extremal coefficients estimator by the fact that it uses extremal coefficients, which are basically stable tail dependence functions.

3. Consider the limiting distribution of \[ (X/t)\mid \max_{v\in V}X_v>t, \qquad t\rightarrow \infty. \] The limit is a so called Huesler-Reiss Pareto distribution with the same matrix \(\Lambda\) as above. For such distributions Engelke and Hitz (2020) have a cliquewise estimator which here it is implemented and called simply Engelke and Hitz estimator. More theoretical background behind the stated limit can be found in Asenova and Segers (2021) and Engelke and Hitz (2020).

4. Consider the limiting distribution of \[ (\ln X_v-\overline{\ln X}, v\in V)\mid \overline{\ln X}>t, \qquad t\rightarrow \infty. \] where \(\overline{\ln X}=(1/|V|)\sum_{v\in V}\ln X_v\). The limit of this vector is also a multivariate Gaussian distribution with mean and covariance matrix, say \(\bar{\Sigma}\), that contain the matrix \(\Lambda(\theta)\) above. We use again sort of MME and MLE to estimate the covariance matrix \(\bar{\Sigma}\), and accordingly \(\theta\), because \(\bar{\Sigma}\) depends on \(\theta\) through \(\Lambda\). This asymptotic result is shown in an unpublished note Segers (2019).

To illustrate the idea: suppose we have the tree with nodes \(V=\{Paris, 2, Meaux, Melun, 5, Nemours, Sens\}\) and edge weights \((t1, \ldots, t6)\) which are aliases for \((\theta_1, \ldots, \theta_6)\).

seg<- graph(c(1,2,
              2,3,
              2,4,
              4,5,
              5,6,
              5,7), directed = FALSE)
name_stat<- c("Paris", "2", "Meaux", "Melun", "5", "Nemours", "Sens")
seg<- set.vertex.attribute(seg, "name", V(seg), name_stat) 
plot(seg, edge.label=c("t1", "t2", "t3", "t4", "t5", "t6"))

We suppose that \((X_{Paris}, X_2)\) have bivariate Huesler-Reiss distribution with parameter \(t1\) or \(\theta_1\). We make the analogous assumptions for all adjacent pair of variables. Then for \(X\) the four asymptotic results above hold and they all depend on the edge weights \((\theta_1, \ldots, \theta_6)\).

To estimate tail dependence in \(X\) we need to estimate \((t1, \ldots, t6)\) (aliases for \(\theta_1, \ldots, \theta_6\)).

For estimation we need to choose between one of the following methods - estimate.MME, estimate.MLE1, estimate.MLE2, estimate.EKS, estimate.EKS_part, estimate.EngHitz, estimate.MMEave, estimate.MLEave.

Block graphs

Let \(G=(V,E)\) be a block graph with node set \(V\) and edge set \(E\). Let \(X\) be a random vector on \(V\) with the following characteristics:

• \(X\) satisfies the global Markov property with respect to the block graph \(G=(V,E)\)

• every distribution of variables belonging to the same maximal clique (or block), say \(C\subset V\), is a multivariate Huesler-Reiss distribution with parameter matrix \(\{\delta^2_{ij}\}, i,j\in C\).

• unit-Pareto marginal distributions.

1. Consider the limiting distribution of \[ \ln (X/X_u)\mid X_u>t, \qquad t\rightarrow\infty. \] It can be shown that the limiting distribution is multivariate Gaussian distribution with mean vector \(\mu_u(\delta)\) and covariance matrix \(\Sigma_{u}(\delta)\) where \(\delta=(\delta_e^2, e\in E)\). One can consult Asenova, Mazo, and Segers (2021) and references therein. Based on this asymptotic result we come up with method of moments type estimator for \(\Sigma_{u}\) and subsequently of \(\delta\).

Similar results as in 2-4 for trees exist for block graphs however we do not implement an estimator based on these results.

To illustrate the idea. Consider the following block graph with three cliques (or blocks) and nine edge parameters: three per each clique.

g<- graph(c(1,3,1,2,2,3,
            3,4,4,5,5,3,
            3,7,3,6,6,7), directed=FALSE)
g<- set.vertex.attribute(g, "name", V(g), letters[1:7])
plot(g, edge.label=c("d1", "d2", "d3", "d4", "d5", "d6", "d7", "d8", "d9"))

We suppose that the subvector \((X_a, X_b, X_c)\) has a Huesler-Reiss distribution with parameter matrix \(\Delta_1\) which is symmetric, with zero diagonal and non-zero parameters \((d2, d1, d3)\) which is alias for \((\delta_{ab}, \delta_{ac}, \delta_{bc})\). Similarly for the other two block vectors \((X_c, X_e, X_d)\) and \((X_c, X_f, X_g)\).

To estimate tail dependence in \(X\) we need to estimate the edge weights (or edge parameters) \((d1, \ldots, d9)\) (aliases for \(\delta_{ab}, (a,b)\in E\)).

For estimation we need to choose method estimate.HRMBG.