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???? ?????Xn,n,...,X1,n?? ??? ?????? ??? F x
F:= sup{x?R,F(x)<1}.???
u? ???? N u:=n? i=11
Xi>u.???
([pn]-pn+ 1)Xn-[pn]-1,n+ (pn-[pn])Xn-[pn],n,???
¯F(x) =
F(x)≈Nun
1 +ˆξˆ
β(x-u)?
-1/ˆξ
ˆxp=u+ˆβˆ
?nN u(1-p)? -1?
ˆxp=Xk+1,n+ˆβˆ
?nk (1-p)? -ˆξ-1? ????p >1-k/n? e n(u) :=1N un i=1[Xi-u]+,??? e(v) := E[vX] =β+ξv1-ξ,???? ?¯F(x)¯
F(Xk+1,n)=L(x)L(Xk+1,n)?
xX k+1,n? -1/ξ
F(x)≈¯F(Xk+1,n)?xX
k+1,n? -1/ξ
F(x) = 1-kn
xX k+1,n? -1/ξHk,????
ˆxHp=Xk+1,n?nk
(1-p)? -ξHk.????
F(x) = 1-?x0x
α,pourx > x0.????
a
0= min{a≥0|Pr[aeρ≥B]>1-α}.????
?????Be-ρ≂ LN(m-μ,⎷s ????? ???? ?? ?????? ??a0? a
0= exp?
(m-μ) +?s
2+σ2Φ-1(1-α)?
1a
0∂a
0∂m
= 1,???? 1a
0∂a
0∂s
=Φ-1(1-α)?1 +σ2/s2.????
ρ=μ+σ0?0+σuN
i=1? i,????
20+λσ2u=σ2.????
H
ξ,β(x) =?
?1-?
1 +ξβ
x? -1/ξ,siξ?= 0,
1-exp(-x/β),siξ= 0.
-β/ξ??ξ <0? E
1 +ξβ
Y? -r =11 +ξr,pourr >-1/ξ, E ln?
1 +ξβ
Y?? k =ξkk!,pourk?N, E ?Y¯Hrξ,β?=β(r+ 1-ξ)(r+ 1),pour (r+ 1)/ξ >0. [ξ-1]?? ???? ?
E[Yr] =βrΓ(ξ-1-r)ξ
r+1Γ(ξ-1+ 1)r!. ????? ???? ????u < yF?
E[Y-u|Y > u] =β+ξu1-ξ,pourβ+uξ >0.
y
F=sup{y?R,F(y)<1}.????
?? ???????f??Y???? f(y) =1β
1 +ξβ
y? -1/ξ-1 ,poury?D(ξ,β).???? lnL(ξ,β;Y1,...,Yn) =-nlnβ-?
1 +1ξ
n? i=1ln?
1 +ξβ
Yi? ??ξ=1n n i=1ln(1 +τYi) =:ˆξ(τ), 1τ =1n 1ξ + 1? n? i=1Y i1+τYi.???? 1τ =1n 1ˆ
ξ(τ)+ 1?
n? i=1Y i1 +τYi.???? n 1/2?
ξn-ξ,ˆβnβ
-1? dN?
0,(1 +ξ)?
1 +ξ-1
-1 2?? , n→ ∞.????
1:= E[Y] =β1-ξ,????
2:= Var[Y] =β2(1-ξ)2(1-2ξ).????
ξ=12
-μ212μ2etβ=μ12
1 +μ21μ
2?
ˆξMM??ˆβMM?
r:= E?Y¯Hrξ,β(Y)?=β(r+ 1)(r+ 1-ξ),pourr= 0;1,????
β=2ω0ω1ω
0-2ω1etξ= 2-ω0ω
0-2ω1.????
ˆξPWM??ˆβPWM?
?ξ <3/2? q p:=H-1
ξ,β(p) =βξ
(1-p)-ξ-1? q p2q
ˆξMQ??
Q p2Q p1=(1-p2)ξ-1(1-p1)ξ-1,???? X
1,n-anb
n→dH,
G(x) = exp?
-(1 +ξx)-1/ξ? ,????1 +ξx≥0,
Λ(x) = exp{-exp(-x)},????x?R.
??H? lim x→∞h(tx)h(x)=tα???? ????t >0. F -1(1-1/x) =x1/αLF-1(x). ???? ?? ?? ??????? ?????? ?? ??????xf(x)/¯F(x)→α?????x→ ∞? ????? ?? lim u→xFsup x>0?|uF(x)-Hξ,β(u)(x)|?= 0. u????? ?????? lim n→∞n¯F(un) =τ?Nun→dP(τ).
1,n,...,x(m)
1,n??X1,n? ?? ??????? ?? ???????
lim u→xFsup x>0?|uF(x)-Hξ,β(u)(x)|?= 0.???? H
Pk=1ln2
lnXk,n-X2k,nX
2k,n-X4k,n,
⎷k ?ξPk-ξ?→dN?
0,ξ2?2ξ+1+ 1?(2(2
ξ-1)ln2)2?
Hk=1k k j=1lnXj,nX k,n, ⎷k ?ξHk-ξ?→dN?0,ξ2?.
DEdHk= 1 +ξ(1)
k-12 (1-?
ξ(1)
k? 2ξ (2) k) )-1 ????ξ(i) k=1k k j=1? lnXj,nX k+1,n? ⎷k ?ξPk-ξ?→dN?0,1 +ξ2?. ⎷k ?ξHk-ξ?→dN?0,ξ2?,???? k ?(n) =? ?1 +n2ξ/(2ξ+1)?(1+ξ)22ξ?
1/(2ξ+1),siξ?]0,1[,
2n2/3,siξ >1.????
quotesdbs_dbs42.pdfusesText_42
Provisions techniques et capital de solvabilité d'une