Rao-Blackwell Theorem & Completeness Definition
Rao-Blackwell Theorem. Let f (x|θ) = f (x1 , . . . , xn |θ) be the joint pdf/pmf of
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(X1 , . . . , Xn ) and S = (S1 , S2 , . . . , Sk ) be sufficient for θ = (θ1 , . . . , θp ) ∈ Θ ⊂ Rp .
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Also let T be any UE of a real-valued γ(θ) and T = E(T |S ) (which does not depend
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on θ since S is sufficient). Then,
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1. T ∗ is a function of S and an UE of γ(θ).
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2. Varθ (T ∗ ) ≤ Varθ (T ), for all θ ∈ Θ.
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3. If, for some θ0 ∈ Θ, it holds that Varθ0 (T ∗ ) = Varθ0 (T ), then Pθ0 (T = T ∗ ) = 1.
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Example: Suppose X1 , X2 are iid Exponential(θ).
(i) Note T = X1 is an UE of θ and Varθ (T ) = Varθ (X1 ) = θ2 .
(ii) Also note that S = X1 + X2 is sufficient by factorization theorem & S is
GAMMA(2,θ)-distributed.
Verify that T ∗ = E(T |S) is a function of S, doesn’t depend on θ, and is unbiased.
Compare Varθ (T ) and Varθ (T ∗ ).
Solution: Given S = s > 0, first find the conditional pdf of X1 |S = s as
 −2 −x /θ −(s−x )/θ
1
θ e 1 e
fX1 ,S (x1 , s|θ)
fX1 ,X2 (x1 , x2 = s − x1 |θ) 
= s−1 if 0 < x1 < s
f (x1 |S = s) =
=
=
θ−2 se−s/θ

fS (s|θ)
fS (s|θ)
0
otherwise
So, given S = s > 0, the distribution of X1 is
Hence, the conditional expectation is E(X1 |S = s) =
Now, treating S as a random variable, we have T ∗ = E(X1 |S) =
1
Notes
• Given an UE T of γ(θ), the theorem shows how to obtain an UE T ∗ that is at
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least as good as T in terms of variance (in fact, better than T unless T = T ∗
with probability 1 for all θ). That is, you can “Rao-Blackwellize” an UE T by
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conditioning on a sufficient statistic S .
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• For finding an UMVUE of γ(θ) we may restrict attention to the class of estimators
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that are functions of a sufficient statistics.
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Proof of Theorem: For any two r.v.s X, Y , it holds that E E(X|Y ) = E(X) and Var(X) =
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E Var(X|Y ) + Var E(X|Y ) . Hence,
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i Eθ (T ∗ ) = Eθ E(T |S ) = Eθ (T ) = γ(θ) for all θ.
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ii Since S is sufficient, the conditional means and variances do not dependent on θ in the following
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(hence no subscript θ on the conditional expectations):
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Varθ (T ) − Varθ (T ) = Varθ E(T |S ) + Eθ Var(T |S ) − Varθ (T ) = Eθ Var(T |S ) ≥ 0
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˜ | {z ˜}
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T∗
for all θ.
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iii Suppose for some θ0 ∈ Θ, Varθ0 (T ∗ ) = Varθ0 (T ) holds. Then by part(ii) above, it holds that
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Eθ0 Var(T |S ) = 0 ⇒ Var(T |S ) = 0 with probability 1 under θ0 ⇒ T = E(T |S) with probability
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1 under θ0 (i.e., conditioned on S , there is no variability in T , so T must equal its conditional
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mean).
“Completeness” Definition: Let f (x|θ) = f (x1 , . . . , xn |θ), θ ∈ Θ ⊂ Rp , be the
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joint pdf/pmf of (X1 , . . . , Xn ) and let fT (t|θ) denote the pdf/pmf of a vector of sta˜ ˜˜
tistics T . Then, T (and/or the family of pdf/pmf FT ≡ {fT (t|θ) : θ ∈ Θ}) is called
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complete if for any real-valued function u(T )
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h
i
(1)
Eθ u(T ) = 0 ∀θ
⇒
Pθ u(T ) = 0 = 1 ∀θ
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T is called bounded complete if (1) holds for all bounded functions u(·).
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