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.

Tangent set of a statistical model at a law

Definition

Fix a model $M$ containing a law $P$ and designate a collection of allowed submodels through $P$ . Each allowed submodel is a one-sided DQM path as in A dominated differentiable-in-quadratic-mean submodel. Its score is the $P$ -almost-sure equivalence class proved unique by A DQM score has mean zero and is unique, and it belongs to $L_{0}^{2} (P)$ (Mean-zero square-integrable functions).

The tangent set $\dot{M} (P)$ is the set of scores of the designated paths. The model specification determines which paths are allowed. In particular, the tangent set need not be a linear space: it may be a cone, meaning it is closed under multiplication by nonnegative real scalars, while failing closure under negative scalars. One-sided restrictions can produce this behavior. The closed linear span of the tangent set is a separate construction.

Depends on

Used by

Closed tangent space Definition
Nuisance tangent space Definition
Pathwise differentiability and parameter influence curves Definition
A boundary score cone differs from its tangent space Example
Bounded mean-zero scores generate the unrestricted tangent space Example

Dependency tree · two levels

15 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

A. W. van der Vaart, Semiparametric Statistics, Lecture 1 §1.2 (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §4 (standard reference, not scraped)