statistique, théorie et gestion de portefeuille - Docs at ISFA

9.1. Les différentes mesures de dépendances extrêmes 275
E.1 Proof of Lemma 1
By definition,
Pr[X > F −1
X
−1
(u), Y > F (u)] =
Let us perform an integration by part :
Defining η = F −1
X
Pr[X > F −1
−1
X (u), Y > F
we obtain the result given in (E.99)
Y
=
∞
F −1
X (u)
∞
F −1
Y (u)
dx
∞
F −1
Y (u)
dy PY (y) · Pɛ(x − αy) (E.106)
dy PY (y) · ¯ Fɛ[F −1
X (u) − αy]. (E.107)
Y (u)] = − ¯ FY (y) · ¯ Fɛ(F −1
X (u) − αy) ∞
+ α
∞
F −1
Y (u)
F −1
Y
(u) +
dy ¯ FY (y) · Pɛ(F −1
X (u) − αy) (E.108)
= (1 − u) ¯ Fɛ(F −1
−1
(u) − αF (u)) +
+ α
∞
F −1
Y (u)
−1
(u) − αF (u) (see equation(E.100)), and dividing each term by
E.2 Derivation of equation (E.102)
Y
X
Y
dy ¯ FY (y) · Pɛ(F −1
X (u) − αy) (E.109)
Pr[Y > F −1
Y (u)] = 1 − u, (E.110)
The factor Y and the idiosyncratic noise ɛ have Student’s distributions with ν degrees of freedom given by
(E.97) and (E.98) respectively. It follows that the survival distributions of Y and ɛ are :
and
¯FY (y) =
ν ν−1
2 Cν
y ν + O(y −(ν+2) ), (E.111)
¯Fɛ(ɛ) = σν ν ν−1
2 Cν
ɛ ν + O(ɛ −(ν+2) ), (E.112)
(E.113)
¯FX(x) = (αν + σ ν ) ν ν−1
2 Cν
x ν + O(x −(ν+2) ). (E.114)
Using the notation (E.101), equation (E.100) can be rewritten as
whose solution for large ˜ Yu (or equivalently as u goes to 1) is
σ
ν1/ν η = α 1 + − 1
α
¯FX(η + α ˜ Yu) = ¯ FY ( ˜ Yu) = 1 − u, (E.115)
˜Yu + O( ˜ Y −1
u ). (E.116)
To obain this equation, we have used the asymptotic expressions of ¯ FX and ¯ FY given in (E.114) and (E.111).
37