PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Then,
U n (θ 0 ; x)
√
ni(θ0 )
−→
D
N(0, 1).
Proof.
U n (θ 0 ; x) = ∂ n
∂θ log ∏
f(x i ; θ)| θ=θ0
=
=
n∑
i=1
i=1
∂
∂θ log f(x i; θ)| θ=θ0
n∑
U i , where U i = ∂ ∂θ log f(x i; θ)| θ=θ0 .
i=1
Since U 1 , U 2 , . . . are i.i.d. with E(U i ) = 0 and Var(U i ) = i(θ), by the Central Limit
Theorem,
∑ n
i=1 U i − nE(U)
√ = U n(θ 0 ; x) −→
√ N(0, 1).
n Var(U) ni(θ0 ) D
Theorem. 2.3.2
Under the above assumptions,
√
ni(θ0 )(ˆθ n − θ 0 ) −→ D
N(0, 1).
Proof. Consider the first order Taylor expansion of U n (ˆθ; x) about θ 0 :
U n (ˆθ n ; x) ≈ U n (θ 0 ; x) + U n(θ ′ 0 ; x)(ˆθ n − θ 0 ).
For large n
=⇒ U n (θ 0 ; x) ≈ −U n(θ ′ 0 ; x)(ˆθ n − θ 0 )
U n (θ 0 ; x)
√
ni(θ0 )
=⇒ −U ′ n(θ 0 ; x)(ˆθ n − θ 0 )
√
ni(θ0 )
−→
D
−→
D
N(0, 1)
N(0, 1).
Now observe that:
−U ′ n(θ 0 ; x)
n
→ i(θ 0 )
as n → ∞
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