Simulation et estimation de quantiles extrêmes et de probabilités d

q
p
p q
q p
10−9
10−12
p
ˆp p
ˆp
q
ˆq q
ˆq
X?
Km

p
pΦ
ˆp
q
•X∈Rd
•Φ : Rd→R
•q X ⇔Φ(X)> q
•µRd
•p X L ∼µP(Φ(X)> q) = p
p q q p
N(Xi)i∈J1,NK
µ p ˆpmc = #{i|Φ(Xi)> q}/N
ˆpmc Var(ˆpmc)/p2∼p→01/(N p)
N≈1/p ≈1012
N(X1
i)i∈J1,NK
µ
m∈N∗(Xm
i)i∈J1,NK
•Lm= mini(Φ(Xm
i))
•Xm+1
i=X?∼ L(X|Φ(X)> Lm) Φ(Xm
i) = Lm
Xm
iΦ(Xm i)> Lm
(Xm
i)i∈J1,NK,m∈N∗
X?(Xm
i)i∈J1,NK
Xm
iΦ(Xm i)
Xm+1
iΦ(Xm+1 i)Xm
i
p q

X?
L(X|Φ(X)> Lm) (Xm
i)i∈J1,NK
X?
L(X|Φ(X)> Lm) (Xm
i)i∈J1,NK
p
q M = max{m|Lm≤q}
ˆp p
M p p M
M
•Φ(X)F:y7→ P(Φ(X)≤y)
•S:y7→ 1−F(y) = P(Φ(X)> y)
•Λ : y7→ −ln(S(y))
(Ei)i∈N∗1
∀m∈N∗,Λ(Lm)L ∼1
N
m
X
i=1
Ei
•
Λ(Φ(X1
i))i∈J1,NKL
∼(Ei)i∈J1,NK
X∼µ, ∀a∈R+,
P(Λ(Φ(X)) ≤a) = P(1 −F(Φ(X)) ≥e−a)
=E(1−F(Φ(X))≥e−a)
F b 1−F(y)≥e−a⇔y≤b F (b)=1−e−a
P(Λ(Φ(X)) ≤a) = E(1−F(Φ(X))≥e−aΦ(X)≥b+1−F(Φ(X))≥e−aΦ(X)<b)
= 0 + E(1 ·Φ(X)<b)
=P(Φ(X)< b)
=F(b)
= 1 −e−a
Λ(Φ) ≥0
Λ(Φ(X1
i))i∈J1,NKL
∼(Ei)i∈J1,NK
•mΛ(Lm)
∗m= 1
Λ
Λ(L1)=Λmin
i∈J1,NK(Φ(X1
i))= min
i∈J1,NKΛ(Φ(X1
i))
Λ(L1)L ∼E1
N

∗m∈N∗
Λ(Lm)L ∼1
N
m
X
j=1
Ej
Lmy > Lm
Λm+1(y) = Λ(y)−Λ(Lm)
Xm+1
j
Λm+1(Φ(Xm+1
j))j∈J1,NKL
∼(Ej)j∈J1,NK
(Li)i∈J1,mK
Λm+1(Lm+1) = Λ(Lm+1)−Λ(Lm)L ∼Em+1
N
Λ(Lm+1)L ∼Λ(Lm) + Em+1
N
Λm+1(Lm+1) Λm(Lm)
Λ(Lm+1)L ∼1
N
m+1
X
j=1
Ej
(Ej)j∈J1,m+1K
MP −Nln(p)
M= max{m|Lm≤q}
= max{m|S(Lm)≥p}
= max{m|Λ(Lm)≤ −ln(p)}
ML ∼ P(−Nln(p))
p→0N≥100 P(−Nln(p))
N(−Nln(p),−Nln(p))
p
ˆp=1−1
NM
ˆp p
S=1−1
Nm
m∈N

∀m∈N,Pˆp=1−1
Nm=pN(−Nln(p))m m!
ˆp p
Var(ˆp) = p2p−1
N−1
•
∀m∈N,Pˆp=1−1
Nm=P1−1 NM
=1−1
Nm
=P(M=m)
=pN(−Nln(p))m
m!
•ˆp p
E(ˆp−p) = E1−1
NM−p
=
+∞
X
m=0 1−1
NmpN(−Nln(p))m m!−p
=pN
+∞
X
m=0 −Nln(p) + Nln(p)
Nm1 m!−p
=pN
+∞
X
m=0 −Nln(p) + ln(p)m1
m!−p
=pNe(1−N) ln(p)−p
= 0
•ˆp
Var(ˆp) = E1−1
NM
−p2
=E1−1
N2M−p2
=
+∞
X
m=0 1−1
N2mpN(−Nln(p))m m!−p2
=pN
+∞
X
m=0 1−1
N2
(−Nln(p))m1
m!−p2
=pNe(−N+2−1
N) ln(p)−p2
=p2p−1
N−1
p→0
Var(ˆp)
p2∼−ln(p) N1 Np ∼Var(ˆpmc) p2
N
X?