statistique, théorie et gestion de portefeuille - Docs at ISFA

9.1. Les différentes mesures de dépendances extrêmes 269
Thus, we have
ρA = ρ
Var(Y | Y ∈ A)
.
Var(X | Y ∈ A)
(B.52)
Using the same method as for the calculation of Cov(X, Y | Y ∈ A), we find
which yields
as asserted in (B.42).
Var(X | Y ∈ A) = E[E(X 2 | Y ) | Y ∈ A)] − E[E(X | Y ) | Y ∈ A)] 2 , (B.53)
= E[E(X 2 | Y ) | Y ∈ A)] − ρ 2 · E[Y | Y ∈ A] 2 , (B.54)
= E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A)] − ρ 2 · Var[Y | Y ∈ A], (B.55)
ρA =
ρ
ρ 2 + E[E(x2 | Y )−ρ 2 Y 2 | Y ∈A]
Var(Y | Y ∈A)
(B.56)
, (B.57)
To go one step further, we have to evaluate the three terms E(Y | Y ∈ A), E(Y 2 | Y ∈ A), and
E[E(X 2 | Y ) | Y ∈ A].
The first one is trivial to calculate :
y∈A dy y · ty(y)
E(Y | Y ∈ A) =
. (B.58)
Pr{Y ∈ A | ν}
The second one gives
so that
y∈A dy y2 · ty(y)
E(Y 2 | Y ∈ A) =
, (B.59)
Pr{Y ∈ A | ν}
⎡
⎢ν
− 1
= ν ⎣
ν − 2 ·
⎤
ν Pr ν−2 Y ∈ A | ν − 2
⎥
− 1⎦
, (B.60)
Pr{Y ∈ A | ν}
⎡
⎢ν
− 1
Var(Y | Y ∈ A) = ν ⎣
ν − 2 ·
ν Pr ν−2 Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
⎤
2 ⎥ y∈A dy y · ty(y)
− 1⎦
−
. (B.61)
Pr{Y ∈ A | ν}
To calculate the third term, we first need to evaluate E(X2 | Y ). Using equation (B.46) and the results given
in (Abramovitz and Stegun 1972), we find
which yields
E(X 2 | Y ) =
= ν + y2
dx
ν + 1
ν + y2 1/2 x2 ν + 1
· tν+1
1 − ρ2 ν + y2
1/2
x − ρy
, (B.62)
1 − ρ2 ν − 1 (1 − ρ2 ) − ρ 2 y 2 , (B.63)
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] = ν
ν − 1 (1 − ρ2 1 − ρ2
) +
ν − 1 E[Y 2 | Y ∈ A] , (B.64)
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