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Rao–Blackwellizing one Bernoulli observation by the sample sum

Example

Work on the finite product sample space ${0, 1}^{n}$ with its discrete sigma-algebra. Let its coordinates $X_{1}, \dots, X_{n}$ be iid Bernoulli $(p)$ with $p \in [0, 1]$ , $n \geq 1$ , and $S = \sum_{i = 1}^{n} X_{i}$ . The unbiased estimator $U = X_{1}$ is Rao–Blackwellized by $S$ to $E_{p} [X_{1} ∣ S] = \frac{S}{n} .$ The statistic $S$ is sufficient: conditional on $S = s$ , the data vector is uniform over the vectors with $s$ ones when $0 < p < 1$ . At $p = 0$ and $p = 1$ , only the fibers $s = 0$ and $s = n$ are supported, respectively, and their conditional laws are point masses at the all-zero and all-one vectors. These conditional laws do not depend on $p$ .

Facts & Assumptions

Given: The finite iid Bernoulli experiment with $p \in [0, 1]$ and $n \geq 1$ .

[F1]

Rao–Blackwellization of an unbiased finite-variance estimator is its conditional expectation given a sufficient statistic. (Rao–Blackwell theorem for scalar finite-variance estimators)

[L1]

Sufficiency is defined by one probability kernel $K$ from the standard-Borel statistic space to the standard-Borel data space satisfying $P_{θ} (A \cap S^{- 1} (B)) = \int_{B} K (s, A) P_{θ}^{S} (d s)$ for every parameter $θ$ and measurable data and statistic events $A, B$ ; the kernel is independent of $θ$ . (Sufficient statistics)

[F2]

$X_{1}$ is a measurable estimator of $p$ , and it is unbiased when its expectation is $p$ . (Statistics and estimators, Unbiased estimators and UMVU estimators)

[F3]

The conditional-expectation notation is an almost-sure class; AC is inherited from its existence convention. (Conditional expectation as an ae class, The Axiom of Choice)

[F4]

A Bernoulli $(p)$ variable takes values $1$ and $0$ with probabilities $p$ and $1 - p$ , and mutual independence of finite-valued variables means their joint cell probabilities factor. (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials, Pairwise and mutual independence of finite-valued random variables)

[F5]

The finite discrete data and statistic spaces are standard Borel: their discrete metrics are complete, and each finite space is separable, so the identity map witnesses the definition. (Standard Borel spaces)

Verification

technique · direct

1.1givenF4L1construct

For $s \in {0, \dots, n}$ let $F_{s} = {x \in {0, 1}^{n} : \sum_{i} x_{i} = s}$ ; it has $(\binom{n}{s}) \geq 1$ elements. Define $K (s, A) = ∣ A \cap F_{s} ∣ / (\binom{n}{s})$ for every data event $A$ . Each section is the uniform probability measure on $F_{s}$ , and every event evaluation is measurable because the statistic space is finite and discrete; thus $K$ is a probability kernel independent of $p$ . By [F4], independence makes the joint law the product of the Bernoulli coordinate masses. For $0 < p < 1$ , each vector in $F_{s}$ has mass $p^{s} (1 - p)^{n - s}$ and $P_{p} (S = s) > 0$ , so $K (s, \cdot)$ is the conditional law given $S = s$ . At $p = 0$ or $p = 1$ , only $s = 0$ or $s = n$ is supported, respectively, and that fiber is a singleton on which the same kernel is the conditional law. Hence $K$ is a common regular conditional law and proves sufficiency by [L1].

2.1F1F2F3F4F5step 1.1algebra∎

For $0 < p < 1$ , the uniform fiber law is invariant under coordinate permutations, so the $n$ conditional means $E_{p} [X_{i} ∣ S = s]$ agree. Their sum is $s$ , hence each is $s / n$ . At $p = 0, 1$ , $X_{1} = S / n$ almost surely, so the same formula holds on the support. Since $0 \leq X_{1} \leq 1$ , $U = X_{1}$ is square-integrable; its Bernoulli law gives $E_{p} X_{1} = p \cdot 1 + (1 - p) \cdot 0 = p$ by [F4]. Both spaces are standard Borel by [F5] and $S$ is sufficient by step 1.1, so [F2] and [F1] identify $S / n$ as the Rao–Blackwellized unbiased estimator; [F3] records its conditional-expectation class convention.

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Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, Theorem 6.1 and Bernoulli example, 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)