Page 27 - Textos de Matemática Vol. 47

MAX-MIN DEPENDENCE COEFFICIENTS 17
Proof. We first deduce the distribution of Wd F j (Yj ). Denoting the copula j=1 j
of X by CX, we have, for each u 2 [0,1],
P
@ j A 1 d
Fj (Yj )  u = CX u , . . . , u = G 1 log u , . . . , 1 log u
0_d 1⇣ 1 1⌘✓1 1◆
j=1 1d = exp ` ( logu) 1,...,( logu) 1
= u`( 1,..., 1). 1d
Then
The next result shows that the max-min coe cient takes into account the taildependencefunctionofallsubvectorsX[j2TIj ofXwithindicesin[j2TIj,
;=6 T✓{1,...,p}.
From the Proposition 2.2 and (2.1) it holds that, for each ; =6 T ✓ {1, . . . , p},
0_d 1Z1( 1 1) E@ F j(Y )A= u` 1 ,..., d ` 1,..., 1 du
jj1d j=1 0
`( 1,..., 1) =1d.
1d
1+`( 1,..., 1) 1d
0 1 `0@X 1 1(Ij),...,X 1 d(Ij)1A __jj
@ j A j2T j2T
E F (Xi) = 0X X 1, (2.5)
j2T j2T
leading to the following relations of the max-min coe cients with the tail de- pendence and the extremal coe cients. For sake of simplicity we denote the above expectation by e (Ij , j 2 T ).
Proposition 2.3. If X has MEV distribution then, for each partition I =
{I1,...,Ip} of D and 2 (0,1)p, it holdsXthat
R(X, , I ) = e (Ij , j 2 {1, . . . , p}) ( 1)|T |+1 e (Ij , j 2 T )
;6=T ✓{1,...,p}
(2.6)
(2.7)
and, for = 1 = (1,...,1),
✏(X) X
( 1)
|T |+1
✏(X[j2T Ij )
R(X,1,I)=1+✏(X)
;6=T ✓{1,...,p}
1+✏(X[ I ). j2T j
⇤
j2T i2Ij 1+`@ 1 1(Ij),..., 1 d(Ij)A jj