PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Proof.
Observe E{U(θ; X)} =
=
=
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
. . . U(θ; x)f(x; θ)dx 1 . . . dx n
−∞
. . .
∫ ∞
−∞
∫ ∞
. . .
−∞
( ∂
∂θ log f(x; θ) )
f(x; θ)dx 1 . . . dx n
⎛ ⎞
∂
f(x; θ)
⎜
⎝
∂θ ⎟
⎠ f(x; θ)dx 1 . . . dx n
f(x; θ)
=
∫ ∞
regularity =⇒ = ∂ ∂θ
= ∂ ∂θ 1
−∞
∫ ∞
∫ ∞
. . .
−∞
−∞
. . .
∂
∂θ f(x; θ)dx 1 . . . dx n
∫ ∞
−∞
f(x; θ)dx 1 . . . dx n
= 0, as required.
To calculate Var{U(θ; X)}, observe
⎧ ⎫
∂
∂ 2 l(θ; x)
= ∂ ⎪⎨ f(x; θ) ⎪⎬
∂θ
∂θ 2 ∂θ ⎪⎩ f(x; θ) ⎪⎭
=
=
=⇒ U 2 (θ; x) =
( )
∂ 2
2 ∂
∂θ f(x; θ)f(x; θ) − 2 ∂θ f(x; θ)
∂ 2
[f(x; θ)] 2
f(x; θ)
∂θ2 − U 2 (θ; x)
f(x; θ)
∂ 2
f(x; θ)
∂θ2 f(x; θ)
− ∂2 l
∂θ 2
=⇒ Var{U(θ; X)} = E{U 2 (θ; X)} (as E(U) = 0)
⎛
∂ 2 ⎞
f(X; θ)
⎜
= E ⎝
∂θ2 ⎟
⎠ + I(θ)
f(X; θ)
(by definition of I(θ)).
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