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Rao–Blackwell theorem for scalar finite-variance estimators

Statement

Assume the Axiom of Choice as in the library conditional-expectation convention. Let $E = ((X, A), Θ, (P_{θ}))$ have standard-Borel data space, let $S$ be standard Borel, and let $S : X \to S$ be a sufficient statistic. Here sufficiency uses a supplied common kernel $K (s, \cdot)$ that is a regular conditional law of the data given $S = s$ under every $P_{θ}$ and does not depend on $θ$ (Sufficient statistics). For Borel $B \subseteq S$ , write $P_{θ}^{S} (B) : = P_{θ} (S^{- 1} (B))$ for the pushforward law of $S$ . Let $ψ : Θ \to R$ and let $U$ be a measurable real estimator with $E_{θ} [U] = ψ (θ)$ and $E_{θ} [U^{2}] < \infty$ for every $θ$ .

There is a single Borel function $g : S \to R$ , independent of $θ$ , such that $g (S)$ is a version of $E_{θ} [U ∣ σ (S)]$ for every $θ$ . It is unbiased, finite-variance, and ${Var}_{θ} (g (S)) \leq {Var}_{θ} (U) (θ \in Θ) .$ For each $θ$ , equality holds exactly when $U$ is $σ (S)$ -measurable $P_{θ}$ -almost surely.

Facts & Assumptions

Given: A standard-Borel experiment and statistic space, a sufficient statistic with the supplied common regular conditional kernel described in the Statement, AC, and an unbiased estimator $U$ with finite second moment under every $P_{θ}$ .

[F1]

A finite-variance unbiased estimator is integrable, has expectation $ψ (θ)$ at every parameter, and has the variance convention stated in the UMVU definition. (Unbiased estimators and UMVU 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)

[F2]

For a probability kernel, integrating a nonnegative measurable function gives a measurable function of the input, with the corresponding iterated-integral identity. (A randomized procedure has a measurable action law)

[F3]

Positive/negative parts, absolute values, and the finite piecewise combinations used to define $g$ preserve measurability. (Closure properties of measurable functions used by the integral)

[F4]

Nonnegative measurable functions admit increasing simple approximations. (Every nonnegative measurable function is the increasing limit of simple measurable functions)

[F5]

The RCD identity for indicators extends to nonnegative increasing limits by monotone convergence and finite linearity on $L^{1}$ . (Monotone convergence for the integral, The Lebesgue integral is linear on $L^{1} (μ)$ )

[F6]

The nonnegative integral is monotone, so $\int u^{\pm} d K \leq \int ∣ u ∣ d K$ . (Monotonicity and nonnegative homogeneity of the nonnegative integral)

[F7]

Every event in $σ (S)$ is $S^{- 1} (B)$ for a Borel statistic-space event $B$ . (The sigma-algebra generated by a function)

[F8]

A conditional-expectation version is measurable, integrable, and has the same event integrals on the conditioning sigma-algebra. (Conditional expectation given a sigma algebra)

[F9]

Under AC, versions with the same event integrals determine one almost-sure class. (Conditional expectation as an ae class)

[F10]

Conditional expectation maps $L^{2}$ into $L^{2}$ . (Conditional lp contraction)

[F11]

Conditional expectation preserves ordinary expectation. (Basic algebra and order properties of conditional expectation)

[F12]

Under AC, the conditional-expectation class of an $L^{2}$ input is its orthogonal projection onto the $L^{2}$ conditioning subspace. (Conditional expectation is the l2 orthogonal projection)

[F13]

The Axiom of Choice is required by the cited conditional-expectation and $L^{2}$ projection interfaces. (The Axiom of Choice)

[F14]

A nonnegative integrable function with zero integral is zero almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

[F15]

A randomized procedure is a parameter-independent probability kernel from an experiment's data space to its measurable action space. (Randomized procedures as probability kernels)

Proof

technique · direct

1.1givenL1F2F3F6F15construct

Write $u$ for the measurable function defining $U$ . For each $θ$ , $P_{θ}^{S}$ is a probability measure: preimages under the measurable map $S$ preserve the empty set, disjoint countable unions, and the whole space. By [L1], $K$ is a parameter-independent probability kernel from the statistic space to the data space, so it is a randomized procedure for the pushforward experiment by [F15]. Thus [F2] and [F3] show that $h_{+} (s) = \int u^{+} d K (s, \cdot)$ , $h_{-} (s) = \int u^{-} d K (s, \cdot)$ , and $r (s) = \int ∣ u ∣ d K (s, \cdot)$ are Borel. Let $g = h_{+} - h_{-}$ where $r < \infty$ and $g = 0$ elsewhere; [F6] makes both terms finite on this Borel set, and [F3] makes $g$ Borel.

2.1givenL1F1F4F5F6F7F8F9F13step 1.1

Fix $θ$ and a Borel statistic event $B$ . For nonnegative simple functions of the data, [L1] and finite linearity give $\int_{B} \int s (x) K (t, d x) P_{θ}^{S} (d t) = E_{θ} [1_{{S \in B}} s (X)]$ . Approximate $u^{+}$ and $u^{-}$ increasingly by simple functions using [F4]; [F5] passes each equality to the limit. Applying the same identity to $∣ u ∣$ with $B = S$ gives $E_{θ} r (S) = E_{θ} ∣ U ∣ < \infty$ . If $N = {r (S) = + \infty}$ had positive probability, then $m 1_{N} \leq r (S)$ for every positive integer $m$ , so [F6] would give $m P_{θ} (N) \leq E_{θ} r (S)$ for all $m$ , a contradiction; thus $r (S) < \infty$ almost surely. By [F6], $h_{+} (S), h_{-} (S) \leq r (S)$ , so $∣ g (S) ∣ \leq 2 r (S)$ and $g (S)$ is integrable. The two finite event-integral identities may therefore be subtracted, giving $E_{θ} [1_{{S \in B}} g (S)] = E_{θ} [1_{{S \in B}} U]$ . By [F7], [F8], and [F9], $g (S)$ is a version of $E_{θ} [U ∣ σ (S)]$ as an almost-sure class for every $θ$ .

3.1F1F10F11F13step 2.1

By [F10], $g (S)$ is square-integrable under each $P_{θ}$ . By [F11], $E_{θ} [g (S)] = E_{θ} [U] = ψ (θ)$ ; hence it is an unbiased finite-variance estimator.

4.1F1F12F13step 3.1algebra

Fix $θ$ . The residual $U - g (S)$ is orthogonal in $L^{2} (P_{θ})$ to every square-integrable $σ (S)$ -measurable function by [F12]. By step 3.1 and $ψ (θ) \in R$ , $g (S) - ψ (θ)$ is such a function. Expanding the centered square gives ${Var}_{θ} (U) = E_{θ} [(U - g (S))^{2}] + {Var}_{θ} (g (S))$ . The first term is nonnegative, proving the variance inequality.

5.1step 4.1F12F14∎

By [F14], equality in step 4.1 holds exactly when $U = g (S)$ $P_{θ}$ -almost surely. Since $g (S)$ is $σ (S)$ -measurable, this is precisely almost-sure $σ (S)$ -measurability: if $U$ has such a representative, its $L^{2}$ class lies in the projection subspace and [F12] fixes it.

Depends on

Used by

Rao–Blackwellizing one Bernoulli observation by the sample sum Example
Lehmann–Scheffé theorem Theorem

Dependency tree · two levels

66 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, Theorem 6.1 and proof, PDF pp. 33–35 (standard reference, not scraped)
Karl B. Gregory, STAT 713 Lecture 6: Best Unbiased Estimators, Rao-Blackwell Theorem, PDF pp. 9–11 (standard reference, not scraped)
Ryan Martin, Lecture Notes on Advanced Statistical Theory, Theorem 5.1 and proof, PDF pp. 102–103 (standard reference, not scraped)