PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
Examples
(1) x 1 , x 2 , . . . , x n i.i.d. N(µ, σ 2 ), σ 2 known.
n∏ 1
Then f(x; µ) = √ e (−1/(2σ2 ))(x i −µ) 2
2πσ
=⇒ S =
i=1
{
}
= (2πσ 2 ) −n/2 exp − 1 n∑
(x
2σ 2 i − µ) 2
i=1
{ ( n∑
)}
= (2πσ 2 ) −n/2 exp − 1
n∑
x 2
2σ 2 i − 2µ x i + nµ 2
i=1
i=1
{ (
)} (
= (2πσ 2 ) −n/2 1
n∑
exp 2µ x
2σ 2 i − nµ 2 exp − 1
2σ 2
= exp
{
(
1
2µ
2σ 2
i=1
)} {
n∑
x i − nµ 2 exp
i=1
n∑
x i is sufficient for µ.
i=1
(2) If x 1 , x 2 , . . . , x n are i.i.d. with
f(x) = exp{A(θ)t(x) + B(θ) + h(x)},
then f(x; θ) = exp
=⇒ S =
{
A(θ)
n∑
t(x i )
i=1
− 1
2σ 2
n∑
i=1
x 2 i
)
}
n∑
x 2 i − n 2 log(2πσ2 )
i=1
} {
n∑
n∑
}
t(x i ) + nB(θ) exp h(x i )
i=1
i=1
is sufficient for θ in the exponential family by the Factorization Theorem.
Theorem. 2.2.5 Rao-Blackwell Theorem
If T is an unbiased estimator for θ and S is a sufficient statistic for θ, then the quantity
T ∗ = E(T |S) is also an unbiased estimator for θ with Var(T ∗ ) ≤ Var(T ). Moreover,
Var(T ∗ ) = Var(T ) iff T ∗ = T with probability 1.
Proof.
(I) Unbiasedness:
θ = E(T ) = E S {E(T |S)} = E(T ∗ )
=⇒ E(T ∗ ) = θ.
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