Page 24 - Textos de Matemática Vol. 47

14 H. FERREIRA
To the best of our knowledge, there is no extremal dependence coe cients for p 3 subvectors X1,...,Xp of a random vector X = (X1,...,Xd) with MEV distribution.
The need to evaluate the strength of dependence among subvectors arises forinstanceintheSsettingofmax-stablerandomfields.Let{Xi}i2R2 bea max-stable random field and I1,...,Ip sets of locations in R2. The joint dis- tribution of Xi,i 2 pj=1 Ij, is a MEV distribution and we want to summarize the dependence among the grouped values {Xi, i 2 Ij } at di↵erent regions Ij , j = 1,...,p. This problem is treated by several authors for two variables Xi and Xj corresponding to two locations i and j (see Naveau et al. (2009) and references therein) and the obtained results are extended for two regions I1 and I2 in Fonseca et al. (2015).
In finance, we are frequently interested in assessing the dependence among several big world markets, considering each one as a random subvector. For an application with grouped financial stock markets see for instance Ferreira and Ferreira (2012a).
We propose to evaluate the degree of dependence among subvectors X1, . . . , Xp of X with MEV distribution by using an expected range, which will be referred as a “max-min coe cient”. It is a summary measure that takes into account the whole group of the extremal coe cients ✏(Xj ) of Xj , j = 1, . . . , p. Our approach is an extension of the modeling for pairwise dependence through- out the madogram (Poncet et al., 2006; Naveau et al., 2009), an extreme-value analogue of the variogram (Cressie, 1993), since it enables to summarize the spatial dependence structure for several locations or regions of locations.
The proposed moment-based dependence tool takes into account the spread and dependence among the subvectors and can be easily estimated.
The paper is organized as follows. We introduce in Section 2 the dependence coe cient which is well defined for any random vector with MEV distribution, is a function of its copula and is invariant with respect permutations of the variables. Its relations with the multivariate tail dependence and the extremal coe cients are presented. Based on the expected range coe cient considered we compare a MEV distribution with others more concordant distributions and state some bounds.
In Section 3, we compute the max-min coe cients for the multivariate mar- ginal distribution of the Multivariate Maxima of Moving Maxima process and the Symmetric Logistic distribution. We refer briefly an estimator for the max- min coe cients and apply it to grouped financial stock markets.