PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
(II) Variance inequality:
Var(T ) = E{Var(T |S)} + Var{E(T |S)}
= E{Var(T |S)} + Var(T ∗ )
≥
Var(T ∗),
since E{Var(T |S)} ≥ 0.
Observe also that Var(T ) = Var(T ∗ )
=⇒ E{Var(T |S)} = 0
=⇒ Var(T |S) = 0 with prob. 1
=⇒ T = E(T |S) with prob. 1.
(III) T ∗ is an estimator:
Since S is sufficient for θ,
T ∗ =
which does not depend on θ.
∫ ∞
−∞
T (x)f(x|s)dx,
Remarks
(1) The Rao-Blackwell Theorem can be used occasionally to construct estimators.
(2) The theoretical importance of this result is to observe that T ∗ will always
depend on x only through S. If T is already a MVUE then T ∗ = T =⇒
MVUE depends on x only through S.
Example
Suppose x 1 , . . . , x n ∼i.i.d. N(µ, σ 2 ), σ 2 known. We want to estimate µ. Take T = x 1 ,
as an unbiased estimator for µ.
n∑
S = x i is a sufficient statistic for µ.
i=1
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