Page 29 - Textos de Matemática Vol. 47

^p
0@\p 1A Yp {M(Ij) > xj}
MAX-MIN DEPENDENCE COEFFICIENTS 19
p
✏(XIj ) ✏(XIj )
j=1 |T|+1 j2T XX X X
ˆ
(c) R(X,1,I)=
Proof. If X has MEV distribution then it is associated (Marshall and Olkin, 1983) and then the variables M (Ij ), j = 1, . . . , p, are also associated (Esary et al., 1967). For these associated variables it holds that
p ( 1) 1+X✏(XIj). 1 + ✏(XIj ) ;6=T ✓{1,...,p} j2T
j=1
P (M(Ij) > xj) P
and 0 1
P (M(Ij) > xj)
j=1
j=1 j=1
^p \p Yp
P (M(Ij)  xj) P @ {M(Ij)  xj}A P (M(Ij)  xj).
j=1 j=1 j=1
By taking Mˆ(Ij) = Wi2Ij Xˆi and M¯(Ij) = Wi2Ij X¯i, j = 1,...,p, we can rewrite the above inequalities as
0\p 1 0\p 1 0\p 1 P@ {M¯(Ij)>xj}A P@ {M(Ij)>xj}A P@ {Mˆ(Ij)>xj}A
j=1 j=1 j=1
and
0\p 1 0\p 1 0\p 1 P@ {M¯(Ij)xj}A P@ {M(Ij)xj}A P@ {Mˆ(Ij)xj}A.
j=1 j=1 j=1
This concordance order implies (Shaked and Shanthikumar, 2007) that
0 0 _p E @ f @
1 1 0 0 _p M¯ ( I j ) A A  E @ f @
1 1 0 0 _p 1 1 M ( I j ) A A  E @ f @ Mˆ ( I j ) A A ,
j=1
and 0 0^p 11 0 0^p 11 0 0^p 11
E @ f @ M¯ ( I j ) A A E @ f @ M ( I j ) A A E @ f @ Mˆ ( I j ) A A j=1 j=1 j=1
for all non-decreasing functions f. The result in (a) follows by taking f = F
and replacing Xi by Xi , i 2 Ij, for each j = 1,...,p. j
j=1
j=1