Page 28 - Textos de Matemática Vol. 47

18 H. FERREIRA
Proof. To obtain the first equality we first apply in (2.3) the relation ^p _ F j (Xi) = X ( 1)|T|+1 _ _ F j (Xi)
j=1i2Ij ;=6 T✓{1,...,p} j2T i2Ij
and then the Proposition 2.1 with (2.5). The statement in (2.7) is a consequence
of (2.6) and (2.1). ⇤ For the particular case of Ij = {j}, j = 1,...,d, we denote R(X, ,I) simply
by R(X, ) and we have, as a consequence of the above result, that `( 1,..., 1) X `( 1 (S),..., 1 (S))
R(X, )= 1 d ( 1)|S|+1 1 1 d d 1+`( 1,..., 1) 1+`( 1 (S),..., 1 (S))
and
1 d;6=S✓{1,...,d} 11 dd
R(X,1) = ✏(X) X ( 1)|S|+1 ✏(XS) . 1 + ✏(X) ;6=S✓{1,...,d} 1 + ✏(XS )
These two relations extend the result of the Proposition 1 in Naveau et al. (2009) and equation (14) in Cooley et al. (2006), where, for d = 2, 1 2 = 1 = 2 (0, 1), we have
1 `(1,1)31 R(X, )= 1
2 1+`(1, 1 ) 2(1+ )(2 ) 1
and R(X, 1) = ✏(X) 1 . ✏(X)+1
Our next step is to compare the value of the max-min coe cient R(X, ,I) with the corresponding coe cient in the two boundary cases of independent or totally dependent XIj , j = 1,...,p.
Proposition 2.4. Let X, Xˆ = (Xˆ1,...,Xˆd) and X¯ = (X¯1,...,X¯d) be vec- tors of unit Fr´echet random variables with MEV distributions such that Xˆ Ij , j = 1,...,p, are independent, X¯Ij , j = 1,...,p, are totally dependent and, for each j = 1, . . . , p, Xˆ Ij , X¯ Ij and XIj are identically distributed. Then, for each 2 (0, 1)d and partition I of D, it holds that
(a) R(X¯, ,I)  R(X, ,I)  R(Xˆ, ,I), ( b ) R ( X¯ , 1 , I ) = 0 ,