Estimation ponctuelle - Mathématiques en ECE 2

\
X
PXX(Pθ)θ∈R
θ
B(n, p)pP(λ)λ . . .
θ
p
X=(0
1
X → B(p)p
X Y
I J R
P([X∈I]∩[Y∈J]) = P(X∈I)P(Y∈J)
(Xn)n∈N
A={i1;. . . ;in} ⊂ NI1, . . . , InR
P n
\
k=1
Xik∈Ik!=
n
Y
k=1
P(Xik∈Ik).

\
(X1, X2, . . . , Xn)
X1,...XpXp+1, . . . , Xn
(X1, X2, X3, X4, X5, X6)Y1=X1+X4Y2=X2+X5Y3=X3+X6
X λ
λX
E(λX) = λE(X).
X1, . . . , Xn
n
P
i=1
Xi
E n
X
i=1
Xi!=
n
X
i=1
E(Xi).
X Y X ≤Y
E(X)≤E(Y).
X Y
XY
E(XY ) = E(X)E(Y).
X Y
X+Y
V(X+Y) = V(X) + V(Y).
n

\
n PXX n X
X
X1, . . . , Xn
x1, . . . , xn
θ Tn=ϕ(X1, X2, . . . , Xn)ϕ
θ n
tn=ϕ(x1, . . . , xn)Tnθ
(x1, . . . , xn)
Tnθ θ
θ
Tnθ
Tnn
N∗
(Tn)n≥1n
n
Tnbθ(Tn) = E(Tn)−θ
(Tn)θ θ
bθ(Tn)=0 E(Tn) = θ Tn
θ
Tnn+∞
θ Tn
TnRθ(Tn) = V(Tn−θ)2
Tnθ θ
TnTn
Rθ(Tn) = bθ(Tn)2+V(Tn)

\
Tn
Rθ=E(Tn−θ)2=E([Tn−E(Tn)] + [E(Tn)−θ])2
Rθ=E[Tn−E(Tn)]2+ [E(Tn)−θ]2+ 2[Tn−E(Tn)][E(Tn)−θ]
Rθ=EV(Tn)+[bθ(Tn)]2+ 2bθ(Tn)[Tn−E(Tn)]
Rθ=V(Tn)+[bθ(Tn)]2+ 2bθ(Tn)E[Tn−E(Tn)]
Rθ=V(Tn)+[bθ(Tn)]2+ 2bθ(Tn)[E(Tn)−E(Tn)]
Rθ=V(Tn)+[bθ(Tn)]2
θ
(Tn)n≥1θ θ
ε > 0
lim
n→+∞P|Tn−θ|> ε= 0.
Tn
(Tn)n≥1θ θ lim
n→+∞Rθ(Tn)=0
(Tn)n≥1

\
p
Tn=1
n
n
P
i=1
Xip
n
P
i=1
Xin p np
np(1 −p)
bp(Tn) = E1
n
n
P
i=1
Xi−p=1
n×np −p= 0
Rp(Tn) = 1
n2Vn
P
i=1
Xi=np(1−p)
n2=p(1−p) n−−−−−→
n→+∞0
Tn
λ
X → P(λ)
Tn=1
n
n
P
i=1
Xiλ
n
P
i=1
Xinλ nλ
nλ
bλ(Tn) = E1
n
n
P
i=1
Xi−λ=1
n×nλ −λ= 0
Rλ(Tn) = 1
n2Vn
P
i=1
Xi=nλ
n2=λ n−−−−−→
n→+∞0
Tn
X
m
Tn=1
n
n
P
i=1
Xim
bm(Tn) = E1
n
n
P
i=1
Xi−m=1
n
n
P
i=1
E(Xi)−m=1
n
n
P
i=1
m−m=1
n×nm −m= 0
V(X)
n2
X
X