Page 30 - Textos de Matemática Vol. 47

20 H. FERREIRA
The equalities in (b) and (c) follow from (2.7) and, in particular for (b), we recall that if X¯ Ij , j = 1, . . . , p, are totally dependent vectors then the copula of X¯ is also the copula of the minimum (Nelsen, 2006). ⇤
The arguments in the proof of (a), of the previous proposition, can be applied to prove that R(X, , I) is a dependence coe cient that decreases with respect to the multivariate concordance ordering.
For the particular case of Ij = {j},Xj = 1, . . . , d, the equality in (c) leads to R(Xˆ,1)= d ( 1)|S|+1 |S|
d + 1 ;=6 S✓{1,...,d} 1 + |S| dXd kd 1
=d+1 ( 1)k+1 dk k+1=d+1, k=1
which extends the already known result for the case of d = 2, where R(Xˆ , 1) = 1.
3
Therefore, if we define
⇢ = 1 d + 1 R ( Xˆ , 1 )
d 1
we have a dependence coe cient with several useful properties: a) several vari- ables can be taken into account; b) it takes values in [0,1] and higher values indicate stronger dependence; c) it is independent of the univariate marginal distributions; d) it can be related with other coe cients in the literature such as the tail dependence and the extremal coe cients; e) it agrees with the concor- dance property for multivariate distributions; f) it has as a particular case the variogram from geostatistics; g) it can be easily implemented and estimated,
as we will see in the next section.
3. Examples and an application
In order to illustrate the previous results, we consider two families of Mul- tivariate Extreme Value distributions and we compute the expressions for R(X, , I), which can be easily implemented.
Example 3.1. If X is the MEV marginal distribution of the Multivariate Max- ima of Moving Maxima processes considered in Smith and Weissman (1996) then `(x1, . . . , xd) = P1l=1 P1k= 1 Wdj=1 xj ↵l,k,j , where the ↵l,k,j are real non- negative constants that sum one. For this tail dependence function we obtain