Statistique paramétrique

X1, . . . , Xn
•Xiα a
•(1 −α)N(0,1)
X1=a1+Y1(1 −1)1∼ B(α)Y1∼ N (0,1)
(δa+λ)⊗n
dPX1,...,Xn
d(δa+λ)⊗n(x1, . . . , xn) =
n
Y
i=1
(αIa(xi) + (1 −α)(1 −Ia(xi))φ(xi))
φN(0,1)
Yi
Yi=1i
0i
Y∗
i
m σ2i
Yi=1Y∗
i≥0
0Y∗
i<0
ΦN(0,1) YiΦ
m σ2
X1X2
λ1λ2X1X2
X∼ E(λ)⇔ ∀x > 0, P (X > x) = exp(−λx)
t
Z
n S1, . . . , Sn
Z1, . . . , Zn
λ1λ2
λ1λ2
θ > 0
fθ(x) = cθ(1 + x)e−θx1[0,+∞[(x)
E(X1)θ
K(θ0|θ1)

a λ a > 0λ > 0 (a, λ)Rγa,λ
γa,λ =λa
Γ(a)e−λxxa−11R+(x) Γ(a) = Z∞
0
e−xxa−1dx.
a > 1 Γ(a) = (a−1)Γ(a−1) n∈N∗Γ(n) = (n−1)!
X(a, λ)E(X) (X)
LX(t) := EetX =λ
λ−ta
t < λ.
X Y (a, λ) (b, λ)
X+Y
Y Y 2
n
χ2n
X S2
X1, . . . , Xnm σ2
X=1
n
n
X
i=1
Xi,Σ2=1
n
n
X
i=1
(Xi−m)2S2=1
n
n
X
i=1
(Xi−X)2.
Σ2σ2Σ2
σ2E(Σ2) = σ2S2
(Xi)N(m, Σ2)
X nΣ2/σ2
X S2
(Xi)X S2
φ Xi
m= 0 E(nS2)σ2E(nS2eitnX ) = (n−1)σ2φn(t)
t φ
φ00
φ−φ0 φ2
=−σ2
φ(0) = 1, φ0(0) = 0.
XiN(0, σ2)ψ(t) = log φ(t)
m= 0