DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Confidence sets and coverage

Definition

Let $E = ((X, A), (P_{θ})_{θ \in Θ})$ be a statistical experiment as in Statistical experiments, and assume $Θ \neq \emptyset$ . A confidence-set procedure is a map $C : X \to P (Θ)$ , where $P (Θ) : = {A : A \subseteq Θ}$ , such that, for every $η \in Θ$ , its membership section $C_{η} : = {x \in X : η \in C (x)}$ belongs to $A$ . Its coverage at $θ$ is $c_{C} (θ) : = P_{θ} (C_{θ})$ , and its confidence level is $\inf_{θ \in Θ} c_{C} (θ)$ , the greatest lower bound of those coverage values. For $0 \leq α \leq 1$ , it has coverage at least $1 - α$ if $P_{θ} (C_{θ}) \geq 1 - α$ for every $θ \in Θ$ .

Only membership sections need to be measurable; no sigma-algebra on the parameter set is assumed. The parameter is fixed in the repeated-sampling coverage probability.

Depends on

Used by

A generalized Bayes credible set can have zero coverage Counterexample
Exact selective confidence set for a parameter functional Definition
A discrete pivot can give conservative coverage Example
Binomial upper confidence limit by lower-tail test inversion Example
Exact unknown-variance normal t interval Example
Known-variance normal mean interval Example
Simultaneous coverage for dependent interval sets Example
Uniform maximum pivot interval Example
Weakly close root laws can still give failed uniform coverage Example
Bonferroni simultaneous coverage Theorem
Coverage from pivot inversion and exact calibration Theorem
Coverage from test inversion Theorem
Exact confidence and prediction intervals in Gaussian regression Theorem
Uniform subsampling coverage under root-law control Theorem

Dependency tree · two levels

16 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

Anna Mikusheva, MIT 14.381, Lecture 11: Confidence Sets, §1, PDF p. 1 (standard reference, not scraped)
Karl B. Gregory, USC STAT 713, Lecture 14: Confidence Intervals and Sets, slides 2–3, PDF pp. 2–4 (standard reference, not scraped)