PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Theorem. 2.2.2 (Cramer-Rao Lower Bound)
If T is an unbiased estimator for θ, then Var(T ) ≥ 1
I(θ) .
Proof.
Observe Cov{T (X), U(θ; X)} = E{T (X)U(θ; X)}
=
=
∫ ∞
−∞
∫ ∞
= ∂ ∂θ
−∞
∫ ∞
∫ ∞
. . . T (x)U(θ; x)f(x; θ)dx 1 . . . dx n
−∞
∫ ∞
. . .
−∞
−∞
. . .
= ∂ E{T (X)}
∂θ
∂
f(x; θ)
T (x) ∂θ
f(x; θ) f(x; θ)dx 1 . . . dx n
∫ ∞
−∞
T (x)f(x; θ)dx 1 . . . dx n
= ∂ ∂θ θ
To summarize, Cov(T, U) = 1.
= 1.
Recall that Cov 2 (T, U) ≤ Var(T ) Var(U) [i.e. |ρ| ≤ 1 and divide both sides by RHS]
=⇒ Var(T ) ≥ Cov2 (T, U)
Var(U)
= 1 , as required.
I(θ)
Example
Suppose X 1 , X 2 , . . . , X n are i.i.d. Po(λ) RV’s, and let ¯X = 1 n
that ¯X is a MVUE for λ.
n∑
X i . We will prove
i=1
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