DISEÑO DE MEZCLA SEGÚN EL MÉTODO DE COMBINACION DE

The definition ofχC in the previous section demonstrates that extremal dependence

in multivariate extremes can have a complicated structure, with only certain subsets of the variables being simultaneously large while other variables are of smaller order. In the radial-angular representation of Section 2.3.6, this corresponds to the spectral measureH placing extremal mass on various faces of the angular simplex Sd−1. The

extremal dependence properties exhibited by a particular set of data should be con- sidered when selecting a model for its extreme values; we should aim to match the extremal dependence structure of the data to the structures that proposed models can capture. Many parametric models for multivariate extremes are only suitable for the asymptotic dependence or asymptotic independence cases, such as the logistic model with α ∈ (0,1) and the multivariate Gaussian, respectively. However, some models, such as the asymmetric logistic model with exponent measure (2.3.5) allow for more complicated extremal dependence structures. In particular, ifP

i∈Cθi,C >0

forC ∈2D\ ∅, the asymmetric logistic distribution places extremal mass on the face of Sd−1 corresponding to the variables {Xi :i ∈C} being simultaneously large while

Figure 2.3.2: Example of a trivariate distribution with mass on five faces of the angular simplex. The data are generated using a multivariate extreme value distribution with an asymmetric logistic model.

Figure 2.3.2 shows an example of data simulated from the asymmetric logistic distribution with the dependence parameters αC = 0.6 for allC ∈2D \ ∅, controlling

how close to the centre of the faces with mass the extremal points lie. The depen- dence structure has been chosen to correspond to limiting mass on two of the vertices, two of the edges and the centre of the unit simplex. In particular, X1 and X3 can both take their largest values independently of the other variables, and the subsets

{X1, X2}, {X2, X3} and {X1, X2, X3} may be simultaneously large. Here, we have taken a sample of size n = 10,000, and the three plots show (W1, W2) |R > r, with

r taken to be the observed 0.9, 0.95 and 0.99 radial quantiles, respectively. The data in Figure 2.3.2 are presented on the equilateral simplex for visual purposes.

As we increase the radial threshold, we observe that, although there are points close to the boundaries of the unit simplex, none of these points lie exactly on the boundary, so simply considering the proportion of points on each face above a high threshold does not reveal the extremal dependence structure. Moreover, there ap- pears to be mass close to all three corners of the unit simplex, when in fact only two of the vertices have positive mass in the limit. To assess the underlying extremal de-

pendence structure of the variables using such data, conditioning onR being above a finite threshold, we need to determine which faces truly represent this limiting depen- dence structure, and which faces appearing to have mass do so as an artefact of lack of convergence at finite levels. We propose methods for determining this structure in Chapters 3 and 4, and introduce some existing methods in the remainder of this section.

Goix et al. (2016) propose a non-parametric simplex-partitioning method for es- timating extremal dependence structures. In this approach, the different regions of the partition are chosen to estimate the various faces of the angular simplex Sd−1, with extremal mass on each of these faces corresponding to a different subset of the variables being simultaneously extreme while the others are of smaller order. Condi- tioning on the radial component being above some high threshold, empirical estimates are obtained for the amount of extremal mass associated with each face, and a sparse representation of the extremal dependence structure is obtained by considering faces where this empirical estimate is sufficiently large. This method is shown to work well in practice, particularly when the asymptotic dependence between variables in the same subsets is strong.

In the method proposed by Goix et al. (2016), the aim is to obtain a representa- tion of the extremal dependence structure that is sparse, i.e., the number of subsets of variables being simultaneously large should be small compared to the dimension of the problem. Chiapino and Sabourin (2017) point out that sparsity may not always be achieved by this method, as too many faces that are similar in some way could be detected. They therefore propose an algorithm that aims to group together nearby faces with extremal mass into feature clusters. This method exploits the graphical structure of clusters, and uses a measure of extremal dependence related toχto group variables that are likely to take their largest values simultaneously. Chiapino et al.

(2019) extend this approach by using the coefficient of tail dependence of Ledford and Tawn (1996), discussed in Section 2.3.9, to assess the extremal dependence of groups of variables. Our approaches in Chapters 3 and 4 exploit a new set of parameters, related to this coefficient of tail dependence, that reveal additional information about the extremal dependence structure.

Extremal dependence structures have also been studied elsewhere in the literature. This includes the factor analysis approach of Kl¨uppelberg et al. (2015) for elliptical copulas; the method of Chautru (2015) which incorporates principal component anal- ysis and clustering techniques; and the Bayesian clustering approach of Vettori et al. (2018).