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.

Statistics and estimators

Definition

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be an experiment (Statistical experiments). A statistic with target measurable space $(Y, B)$ is a measurable map $T : (X, A) \to (Y, B)$ , where measurability means that $T^{- 1} (B) \in A$ for every $B \in B$ (A measurable function between measurable spaces). If $ψ : Θ \to Y$ is an estimand (Estimands on a statistical model) and its target set $Y$ is equipped with $B$ , a statistic $\hat{ψ}$ with this same target space is an estimator of $ψ$ . The value $\hat{ψ} (x)$ is the estimate at observed data $x$ . Giving $Y$ a measurable structure is an additional choice; this definition does not assert that $ψ$ is measurable as a function of the parameter.

Depends on

Used by

Aalen–Johansen cumulative-incidence estimator Algorithm
Delete-d jackknife replicates and their uniform average Algorithm
Size-n bootstrap from the empirical law Algorithm
Split conformal prediction set Algorithm
Uniform subsampling without replacement Algorithm
Asymptotic normality of the Cox partial-likelihood estimator Corollary
The logrank ordering gives a valid Fisher randomization p-value Corollary
A curved normal family can make a sufficient statistic incomplete Counterexample
A Gaussian synthetic likelihood can change an exact posterior Counterexample
A nonconstant summary can lose an identifiable parameter Counterexample
A nonmeasurable rule is not a statistic or test Counterexample
An identifiable model can have persistent ABC summary distortion Counterexample
Positive ABC tolerance changes an identity-summary target Counterexample
Uniform moving support invalidates the naive score bound Counterexample
Ancillary statistics Definition
Approximate Bayesian computation and the tolerance-conditioned target Definition
Bias and mean squared error Definition
Binding function and indirect-inference target Definition
Complete statistics Definition
Conformal scores and rank p-values Definition
Covariance and correlation principal component analysis Definition
Cross-validation splits and training-size risk targets Definition
Density targets and the kernel density estimator Definition
Exponential families and full rank Definition
Families with a monotone likelihood ratio Definition
Fisher sharp null and predeclared test orderings Definition
Generalized least-squares estimator for known covariance Definition
Ideal and finite-simulation synthetic likelihood Definition
IID empirical bootstrap sample and mean statistic Definition
Linearly estimable contrasts in fixed-design regression Definition
Local-polynomial estimator Definition
m-out-of-n bootstrap and subsampling Definition
Minimal sufficient statistics Definition
Nadaraya–Watson estimator Definition
Nelson–Aalen, Kaplan–Meier, and Greenwood estimators Definition
Pointwise squared-error risk for regression Definition
Scientific statistical model contracts Definition
Studentized two-sample U-statistic and permutation distribution Definition
Sufficient statistics Definition
The iid multivariate-normal sample and covariance conventions Definition

…and 45 more results.

Dependency tree · two levels

8 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, PDF pp. 100–102 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §2, PDF pp. 2–3 (standard reference, not scraped)