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9.2. Estimation du coefficient de dépendance de queue 323
H3: There is two constant (l1, l2) ∈ R+ × R+, such that
F1
lim
u→1
−1 (u)
FY −1 (u) = l1,
F2
and lim
u→1
−1 (u)
FY −1 (u) = l2, (66)
then, the coefficient of (upper) tail dependence of (X, Y ) is given by
A.2.2 Proof
λ =
∞
maxl 1
β1 , l 2
β2 dx f(x). (67)
We first give a general expression for the probability for X to be larger than F −1
X (u) knowing that
(u) :
Y is larger than F −1
Y
Lemma 3
The probability that X is larger than F −1
X
F −1
Y (u)
1 − u
dx PY
−1
(u) knowing that Y is larger than F (u) is given by :
Pr X > F −1
−1
X (u)|Y > FY (u) =
−1
FY (u) x · ¯
Fε1,ε2 F1 −1 (u) − β1F −1
Y (u) x, F2 −1 (u) − β1F −1
Y (u) x . (68)
Proof : The proof is the same for lemma 1.
Let us now define the function
F −1
Y (u)
fu(x) =
1 − u PY (F −1
We can state the following result
Lemma 4
Under assumption H1 and H3, for all x ∈ [1, ∞),
almost everywhere, as u goes to 1.
Y (u) x) · ¯ Fε1,ε2 [F1 −1 (u) − β1F −1
Y (u) x, F2 −1 (u) − β2F −1
Y
fu(x) −→ 1x>maxl 1
β1 , l 2
Proof: Applying the assumption H3, we have
and
lim
u→1 F1 −1 (u) − β1F −1
Y
lim
u→1 F2 −1 (u) − β2F −1
Y
(u) x = lim
=
u→1
(u) x = lim
=
Y
(u) x] . (69)
β2 · f(x), (70)
β1F −1
Y (u)
−∞ if x > l1
β1 ,
∞ if x < l1
β1 ,
u→1
β2F −1
Y (u)
−∞ if x > l2
β2 ,
∞ if x < l2
β2 ,
24
F1 −1 (u)
β1F −1 − x
Y (u)
F2 −1 (u)
β2F −1 − x
Y (u)
(71)
(72)
(73)
(74)
(75)
(76)