Estimation ponctuelle I Estimateur et estimation

θ
θ
X θ
θ
θ n X
(x1,··· , xn)
X(Ω,T, P )
n X (X1, X2, . . . , Xn)
(Ω,T, P )X
X
(X1, X2, . . . , Xn)n X
θ
θ Tn
X1, X2, . . . , Xn
Tn=ϕn(X1, X2, . . . , Xn)
(x1, x2, . . . , xn)ϕn(x1, x2, . . . , xn)
θ Tn

Tnθ
Tnθ,
Tn
b(Tn) = E(Tn−θ) =
Tnθ
Tnθ
Tnθ
θ
θ
(Tn)n∈N∗θ(Tn)n∈N∗
θ
(Tn)θ
Tn
TnSnθ Tn
Sn

Tnθ Tnθ
Tn
E(Tn−θ)2
Tnθ
E(Tn−θ)2=
E(Tn−θ)2=E(T2
n−2θ Tn+θ2)
=E(T2
n)−2θE(Tn) + θ2E(T2 n) = V(Tn) + E(Tn)2
=V(Tn)+(E(Tn))2−2θE(Tn) + θ2
=V(Tn)+(E(Tn)−θ)2=V(Tn)+(b(Tn))2b(Tn) =
Tnθ, E(Tn−θ)2=
(X1, X2, . . . , Xn)n X
X µ σ2
X(x1, x2, . . . , xn)xe
Xn=1
n(X1+X2+··· +Xn)
Xn
Xn=

E(Xn) = 1
n
n
X
i=1
E(Xi) =
V(Xn) = 1
n2V n
X
i=1
Xi!=
E(Xn) = V(Xn) =
n
n X1, X2, . . . , Xn
µ σ2
n>30 Xn
X
Xnn < 30
X
X →
XiN(µ , σ)
Sn=X1+X2+···+Xn
Xn=Sn
n→
Xn−µ
σ/√n

n < 30 Xn→
xen
bµ µ
p p
X
P(X= 1) = . . . P (X= 0) = . . .
X →
E(X) = V(X) =
n(X1, X2, . . . , Xn)n X
Sn=
n
X
i=1
Xi. . .
Fn=1
n
n
X
i=1
Xi
n
Fn
E(Fn) = V(Fn) =
Sn=
n
X
i=1
Xi→
∀k∈J0, nK, P Fn=k
n=
Fnn
n > 30
n p >15
np(1 −p)>5
Fn. . .