ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

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A batch-size sequence fails the OBM consistency conditions

Example

Set $δ = 1$ and $b_{n} = ⌊ n^{1 / 2} ⌋$ for $n \geq 2$ . This sequence fails one Flegal–Jones batch-size condition, $n^{2 / (2 + δ)} (\log n)^{3} / b_{n} \to 0$ . The example establishes inapplicability of this sufficient condition only; it does not imply that the OBM estimator is inconsistent.

Facts & Assumptions

Given: The sequence $b_{n} = ⌊ n^{1 / 2} ⌋$ and the lower-growth batch-size premise retained in the corrected Flegal–Jones OBM contract.

[F1]

The Flegal–Jones batch-size contract for exponent $δ$ includes the lower-growth limit $n^{2 / (2 + δ)} (\log n)^{3} / b_{n} \to 0$ . (Flegal–Jones overlapping-batch size conditions)

[F2]

The OBM estimator uses overlapping batch means of integer length $b_{n} < n$ . (Overlapping batch means for MCMC standard errors)

Verification

1.1F1givenalgebra

For $n \geq 2$ , $n^{1 / 2} - 1 < b_{n} \leq n^{1 / 2}$ , so $b_{n} / n^{1 / 2} \to 1$ . Thus $n^{2 / 3} (\log n)^{3} / b_{n} \sim n^{1 / 6} (\log n)^{3} \to \infty$ , which violates the required limit in [F1] at $δ = 1$ .

2.1F2step 1.1givenalgebra∎

Since $1 \leq b_{n} \leq n^{1 / 2} < n$ for $n \geq 2$ , the finite OBM procedure in [F2] is defined. Step 1.1 establishes failure of one source sufficient condition only, not inconsistency of the estimator.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

James M. Flegal and Galin L. Jones, Batch means and spectral variance estimators in Markov chain Monte Carlo, Theorem 2 and Corollary 1 (standard reference, not scraped)