PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
I(λ) =
( ) ∂ 2 l
−E
∂λ 2
=⇒ I(λ) =
( ) n ¯X
E
λ 2
= n E( ¯X)
λ2 = nλ
λ 2
= n λ .
(3) Finally, observe that Var( ¯X) = λ n = 1
I(λ) .
By Theorem 2.2.2, any unbiased estimator T for λ must have
Var(T ) ≥ 1
I(λ) = λ n
=⇒ ¯X is a MVUE.
Theorem. 2.2.3
The unbiased estimator T (x) can achieve the Cramer-Rao Lower Bound only if the
joint PDF/probability function has the form:
f(x; θ) = exp{A(θ)T (x) + B(θ) + h(x)},
where A, B are functions such that θ = −B′ (θ)
, and h is some function of x.
A ′ (θ)
Proof. Recall from the proof of Theorem 2.2.2 that the bound arises from the inequality
where U(θ; x) = ∂l
∂θ .
Cor 2 (T (X), U(θ; X)) ≤ 1,
Moreover, it is easy to see that the Cramer-Rao Lower Bound (CRLB) can be achieved
only when
Cor 2 {T (X), U(θ; X)} = 1
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