TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Unbiased variance estimate from independent computation replicates

Statement

Let $Z_{1}, \dots, Z_{R}$ be IID real random variables with mean $μ$ and finite variance $τ^{2}$ , where $R \geq 2$ , and let ${\overline{Z}}_{R} = R^{- 1} \sum_{r = 1}^{R} Z_{r}$ . Then ${\hat{V}}_{R} : = \frac{1}{R (R - 1)} \sum_{r = 1}^{R} (Z_{r} - {\overline{Z}}_{R})^{2}$ is unbiased for $Var ({\overline{Z}}_{R}) = τ^{2} / R$ . The probability law may be the computation-randomization law conditional on fixed observed data; in that use, this estimate concerns computation variability only.

Facts & Assumptions

Given: IID real random variables $Z_{1}, \dots, Z_{R}$ with finite mean $μ$ and variance $τ^{2}$ , and an integer $R \geq 2$ . Prove the stated unbiasedness.

[F1]

The variance of $Z_{r}$ is $E [(Z_{r} - E Z_{r})^{2}]$ . (Moments, variance, and covariance on a probability space)

[F2]

Expectations of products of independent integrable real random variables factor. (Expectations factor over finite products of independent random variables)

[F3]

IID means identically distributed and independent; no moment assumption is implicit in that definition. (Identical distribution and IID families)

Proof

1.1givenF1F3algebra

By [F3] and the given moment assumptions, every $Z_{r}$ has $E Z_{r} = μ$ and $E Z_{r}^{2} = μ^{2} + τ^{2}$ .

1.2givenF2F3

For $r \neq s$ , independence and [F2] give $E [Z_{r} Z_{s}] = E Z_{r} E Z_{s} = μ^{2}$ .

1.3givenalgebra

Expanding the finite square gives $\sum_{r} (Z_{r} - {\overline{Z}}_{R})^{2} = \sum_{r} Z_{r}^{2} - R {\overline{Z}}_{R}^{2}$ , since $\sum_{r} Z_{r} = R {\overline{Z}}_{R}$ .

2.1step 1.1step 1.2F1algebra

Expanding ${\overline{Z}}_{R}^{2}$ and using steps 1.1–1.2 gives $E [{\overline{Z}}_{R}^{2}] = R^{- 2} {R (μ^{2} + τ^{2}) + R (R - 1) μ^{2}} = μ^{2} + τ^{2} / R$ ; hence $Var ({\overline{Z}}_{R}) = τ^{2} / R$ .

3.1step 1.1step 1.3step 2.1algebra

Taking expectations in step 1.3 and using steps 1.1 and 2.1 gives $E \sum_{r} (Z_{r} - {\overline{Z}}_{R})^{2} = R (μ^{2} + τ^{2}) - R (μ^{2} + τ^{2} / R) = (R - 1) τ^{2}$ .

4.1step 3.1step 2.1algebra∎

Divide step 3.1 by $R (R - 1)$ to obtain $E {\hat{V}}_{R} = τ^{2} / R = Var ({\overline{Z}}_{R})$ , as required.

Depends on

Used by

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Sources

Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §17.1 (standard reference, not scraped)
Anthony Davison and David Hinkley, Bootstrap Methods and Their Application, Chapter 2 §2.5.2 (standard reference, not scraped)