Optimality

Copulas, information, dependence and decoupling 195
the density of a dependent sample) φ2 = I(f0, f; Φ1), where Φ1(x) = x; the relative
entropy satisfies δ = I(f0, f; Φ2), where Φ2(x) = log(x); and the multivariate
divergence measures satisfy D ψ
X1,...,Xn = I(f0, f,Φ3), where Φ3(x) = ψ(x)/x.
If gi1,...,ic(xi1, . . . , xic) are functions corresponding to Theorem 2.1 and
Remark 2.2, then from Theorem 4.1 it follows that the measures δX1,...,Xn, φ2 X1,...,Xn ,
D ψ
X1,...,Xn
written as
(7.1)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
, ρ(q)
X1,...,Xn (in particular, 2H2 X1,...,Xn for q = 1/2) and I(f0, f; Φ) can be
δX1,...,Xn = E log (1 + Un(X1, . . . , Xn))
= E (1 + Un(ξ1, . . . , ξn))log(1 + Un(ξ1, . . . , ξn)) ,
φ 2 X1,...,Xn = E (1+Un(ξ1, . . . , ξn)) 2 − 1
= EU 2 n(ξ1, . . . , ξn) = EUn(X1, . . . , Xn),
D ψ
X1,...,Xn = Eψ (1 + Un(ξ1, . . . , ξn)) ,
ρ (q)
X1,...,Xn = (1/(1−q))(1−E(1 + Un(ξ1, . . . , ξn)) q ),
2H 2 X1,...,Xn = 1/2(1−E(1 + Un(ξ1, . . . , ξn)) 1/2 ),
I(f0, f; Φ) = EΦ(1 + Un(ξ1, . . . , ξn)) (1 + Un(ξ1, . . . , ξn)),
where Un(x1, . . . , xn) is as defined by (4.1).
From (7.2) it follows that the following formula that gives an expansion for
φ 2 X1,...,Xn in terms of the “canonical” functions g holds: φ 2 X1,...,Xn =
�n �
c=2 1≤i1