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Lehmann–Scheffé theorem

Statement

Assume AC as in the library conditional-expectation convention. Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment with standard-Borel data space. Let $S : (X, A) \to (S, B)$ be a statistic with standard-Borel target, sufficient under the common conditional-law interface of Sufficient statistics and complete under Complete statistics. Let $ψ : Θ \to R$ be a real-valued estimand and let $g : S \to R$ be Borel, with $g (S)$ unbiased for $ψ$ and of finite variance under every $P_{θ}$ . Then $g (S)$ is the unique UMVU estimator of $ψ$ : for every finite-variance unbiased estimator $U$ , ${Var}_{θ} (g (S)) \leq {Var}_{θ} (U) (θ \in Θ),$ and any other UMVU estimator equals $g (S)$ $P_{θ}$ -almost surely for each $θ$ .

Facts & Assumptions

Given: AC, a standard-Borel experiment, a complete sufficient statistic $S$ , and a finite-variance unbiased estimator $g (S)$ of a real estimand $ψ$ .

[F1]

UMVU compares the variances of unbiased finite-variance estimators at every parameter. (Unbiased estimators and UMVU estimators)

[F2]

For any unbiased finite-variance estimator $U$ , Rao–Blackwellization by $S$ is an unbiased finite-variance function of $S$ , has no larger variance, and equality at $θ$ holds exactly when $U$ is $σ (S)$ -measurable $P_{θ}$ -almost surely. (Rao–Blackwell theorem for scalar finite-variance estimators)

[L1]

Sufficiency supplies one probability kernel $K$ from the standard-Borel statistic space to the standard-Borel data space such that, for every parameter $θ$ , data event $A$ , and statistic event $B$ , $P_{θ} (A \cap S^{- 1} (B)) = \int_{B} K (s, A) P_{θ}^{S} (d s)$ . The same kernel is fixed independently of $θ$ . (Sufficient statistics)

[L2]

Completeness says that for every measurable real-valued $h (S)$ integrable under every $P_{θ}$ , if $E_{θ} h (S) = 0$ for every $θ$ , then $h (S) = 0$ $P_{θ}$ -almost surely for every $θ$ . (Complete statistics)

[F3]

The conditional-expectation results used by Rao–Blackwell assume AC, which is stated in that theorem and carried here. (The Axiom of Choice)

[F4]

An integrable $G$ -measurable function is its own conditional-expectation version given $G$ . An integrable function equal to it almost surely has the same conditional-expectation class, and versions determine one almost-sure class. (Conditional expectation given a sigma algebra, Conditional expectation as an ae class)

Proof

technique · direct

1.1F1F2L1F3

Let $U$ be any finite-variance unbiased estimator of $ψ$ . The sufficient-statistic hypothesis [L1] permits applying [F2]. Under the stated AC branch [F3], there is one Borel function $u_{S} : S \to R$ , independent of $θ$ , such that $u_{S} (S)$ is a version of $E_{θ} [U ∣ σ (S)]$ for every $θ$ . Rao–Blackwell [F2] makes it unbiased and finite-variance and gives ${Var}_{θ} (u_{S} (S)) \leq {Var}_{θ} (U)$ for every $θ$ .

2.1F1F2L2step 1.1

The difference $u_{S} (S) - g (S)$ is integrable, is a function of $S$ , and has expectation zero for every $θ$ . Completeness [L2] therefore gives $u_{S} (S) = g (S)$ $P_{θ}$ -almost surely for every $θ$ . Substitution into step 1.1 proves ${Var}_{θ} (g (S)) \leq {Var}_{θ} (U)$ for every $θ$ , so $g (S)$ is UMVU.

3.1F1F2F4step 1.1step 2.1∎

If $V$ is another UMVU estimator, applying steps 1.1 and 2.1 to $V$ gives its Rao–Blackwellized function $v_{S} (S) = g (S)$ almost surely for each $θ$ . Since both $V$ and $g (S)$ are UMVU, their variances agree at each $θ$ . The equality condition in [F2] makes $V$ $σ (S)$ -measurable $P_{θ}$ -almost surely; by [F4], such a representative is its own conditional expectation, so uniqueness gives $V = v_{S} (S) = g (S)$ almost surely.

Depends on

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Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, Theorem 6.5 and proof, PDF pp. 37–38 (standard reference, not scraped)
Karl B. Gregory, STAT 713 Lecture 6: Best Unbiased Estimators, Rao-Blackwell Theorem, PDF pp. 15–16 (standard reference, not scraped)
Jun Shao, STAT 709 Lecture 15: UMVUE, PDF pp. 1–4 (standard reference, not scraped)