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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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Pointwise risk is variance plus squared bias

Statement

Assume AC as used to select the regression target version. Fix a model parameter $θ$ and point $x$ . Let $T_{x}$ be a real estimator of the finite target $ψ = m_{θ} (x)$ with $E_{θ} [T_{x}^{2}] < \infty$ . Define its bias by $b_{θ} (x) = E_{θ} [T_{x}] - m_{θ} (x)$ . Its pointwise squared-error risk from Pointwise squared-error risk for regression satisfies $R_{x} (θ, T_{x}) = {Var}_{θ} (T_{x}) + b_{θ} (x)^{2} .$

Facts & Assumptions

Given: The target, estimator, finite second moment, and AC hypothesis in the statement.

[F1]

Pointwise squared-error risk is $R_{x} (θ, T_{x}) = \int (T_{x} (z) - m_{θ} (x))^{2} P_{θ} (d z)$ . Pointwise squared-error risk for regression.

[F2]

For square-integrable $T_{x}$ , ${Var}_{θ} (T_{x}) = E_{θ} [(T_{x} - E_{θ} T_{x})^{2}]$ . Moments, variance, and covariance on a probability space.

[A1]

AC is used only through the selected real version of $m_{θ} (x)$ in Regression target and chosen version; the algebra below is choice-free.

Proof

technique · direct expansion

1.1givenF2A1algebra

Write $T_{x} - m_{θ} (x) = (T_{x} - E_{θ} T_{x}) + b_{θ} (x)$ . Squaring and taking expectations gives $E_{θ} [(T_{x} - m_{θ} (x))^{2}] = E_{θ} [(T_{x} - E_{θ} T_{x})^{2}] + b_{θ} (x)^{2}$ , because $E_{θ} [T_{x} - E_{θ} T_{x}] = 0$ .

2.1givenF1F2step 1.1∎

By [F1] the left side of step 1.1 is the pointwise risk, and by [F2] its first term on the right is the variance. This is the asserted identity.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.1 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§4.1–4.2 (standard reference, not scraped)