Rao-Blackwell Theorem & Completeness Definition Rao-Blackwell Theorem. Let f (x|θ) = f (x1 , . . . , xn |θ) be the joint pdf/pmf of ˜˜ ˜ (X1 , . . . , Xn ) and S = (S1 , S2 , . . . , Sk ) be sufficient for θ = (θ1 , . . . , θp ) ∈ Θ ⊂ Rp . ˜ ˜ ∗ Also let T be any UE of a real-valued γ(θ) and T = E(T |S ) (which does not depend ˜ ˜ on θ since S is sufficient). Then, ˜ ˜ 1. T ∗ is a function of S and an UE of γ(θ). ˜ ˜ 2. Varθ (T ∗ ) ≤ Varθ (T ), for all θ ∈ Θ. ˜ ˜ ˜ 3. If, for some θ0 ∈ Θ, it holds that Varθ0 (T ∗ ) = Varθ0 (T ), then Pθ0 (T = T ∗ ) = 1. ˜ ˜ ˜ ˜ 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 ˜ 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 ˜ conditioning on a sufficient statistic S . ˜ • For finding an UMVUE of γ(θ) we may restrict attention to the class of estimators ˜ that are functions of a sufficient statistic. ¡ ¢ Proof of Theorem: For any two r.v.s X, Y , it holds that E E(X|Y ) = E(X) and Var(X) = ¡ ¢ ¡ ¢ E Var(X|Y ) + Var E(X|Y ) . Hence, ¡ ¢ i Eθ (T ∗ ) = Eθ E(T |S ) = Eθ (T ) = γ(θ) for all θ. ˜ ˜ ˜ ˜ ˜ ˜ ii Since S is sufficient, the conditional means and variances do not depend on θ in the following ˜ ˜ (hence no subscript θ on the conditional expectations): ˜ · ³ ´ ³ ´¸ ³ ´ ∗ Varθ (T ) − Varθ (T ) = Varθ E(T |S ) + Eθ Var(T |S ) − Varθ (T ∗ ) = Eθ Var(T |S ) ≥ 0 ˜ ˜ ˜ ˜ ˜ | {z ˜} ˜ ˜ ˜ T∗ for all θ. ˜ iii Suppose for some θ0 ∈ Θ, Varθ (T ∗ ) = Varθ (T ) holds. Then by part(ii) above, it holds that 0 0 ˜ ˜ ˜ ¡ ¢ Eθ Var(T |S ) = 0 ⇒ Var(T |S ) = 0 with probability 1 under θ0 ⇒ T = E(T |S) with probability 0 ˜ ˜ ˜ ˜ 1 under θ0 (i.e., conditioned on S , there’s no variability in T , so T equals its conditional mean). ˜ ˜ “Completeness” Definition: Let f (x|θ) = f (x1 , . . . , xn |θ), θ ∈ Θ ⊂ Rp , be the ˜˜ ˜ ˜ joint pdf/pmf of (X1 , . . . , Xn ) and let fT (t|θ) denote the pdf/pmf of a vector of ˜ ˜˜ statistics T . Then, ˜ (i) T (and/or the family of pdf/pmf FT ≡ {fT (t|θ) : θ ∈ Θ}) is called complete if ˜ ˜ ˜ ˜˜ ˜ for any real-valued function u(T ) ˜ h i (1) Eθ u(T ) = 0 ∀θ ⇒ Pθ u(T ) = 0 = 1 ∀θ ˜ ˜ ˜ ˜ ˜ ˜ (ii) T is called bounded complete if (1) holds for all bounded functions u(·). ˜ 2