Page 26 - Textos de Matemática Vol. 47

16 H. FERREIRA partition I = {I1,...,Ip} of D, we define
R(X, , I ) = E We remark the relations
F j (M (Ij ))
^p j=1
F j (M (Ij ))
.
R(X, , I ) = E 0@ _
{i,j }⇢{1,...,p}
0@ _p j=1
1A
F j (M (Ij )) F i (M (Ii )) 1A (2.2)
and
R(X, ,I)=E0@_p _ F j(Xi) ^p _ F j(Xi)1A. (2.3)
j=1 i2Ij j=1 i2Ij
By taking p = 2 in (2.2), we find in 1 R(X, , I) the generalized madogram
2
introduced in Fonseca et al. (2015), which in turns is the the -madogram (Naveuetal.,2009)whend=2=pand1 2 = 1 2(0,1).
The max-min coe cient and the generalized madograms for pairs of sets Ii and Ij can be related throughout
R(X, ,I) _ R((XIi,XIj ),( i, j),{Ii,Ij}). {i,j }⇢{1,...,p}
We first present a key result that relates the expectation of Wdj=1 F j (Xj ) with the tail dependence function of G, which enables the derivation of the main properties of R(X, ,I). The result also points out that in this work we can assume that the MEV distributions have unit Fr´echet margins without loss of generality.
Proposition 2.2. Let X be a random vector with MEV distribution G, unit Fr´echet F margins and tail dependence function `. If Y has MEV distribution with marginal distributions Fj, j = 1,...,d, and the same copula as X, then for each 2 (0,1)d, it holds that
0 _d 1 0 _d 1 ` ( 1 , . . . , 1 ) E @ F j (Yj )A = E @ F j (Xj )A = 1 d
. (2.4)
j 1+`( 1,..., 1)
j=1 j=1
1 d