TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Sufficiency makes the conditional rejection probability nuisance-free

Statement

Let $(X, A)$ and $(S, S)$ be standard Borel spaces as in Standard Borel spaces. Fix Polish topologies realizing these specified Borel sigma-algebras, as the standard-Borel definition permits. Let $Θ_{0}, Θ_{1}$ be nonempty disjoint parameter subsets, and write ${P_{η} : η \in Θ_{0}}$ for the null subfamily. Let $S : X \to S$ be measurable. Suppose there is one probability kernel $K (s, \cdot)$ such that, for every $η \in Θ_{0}$ , $K (S, \cdot)$ is a regular conditional distribution of the full data $X$ given $σ (S)$ under $P_{η}$ . This is the common-kernel premise of sufficiency for the null subfamily in Sufficient statistics. For any measurable $φ : X \to [0, 1]$ , define $q (s) : = \int_{X} φ (x) K (s, d x) .$ Then $q$ is measurable and $q (S)$ is a version of the conditional rejection probability of $φ$ given $S$ under every $P_{η}$ . The same function $q$ works for every nuisance value, with conditional-expectation identities interpreted almost surely under each corresponding law.

Facts & Assumptions

Given: The spaces, null family, statistic, common probability kernel and bounded test function in the Statement.

[A1]

For every $η \in Θ_{0}$ , the same kernel $K (S, \cdot)$ satisfies the conditional event-integral identities for $X$ given $S$ under $P_{η}$ , which is the common-kernel sufficiency condition in Sufficient statistics.

[F1]

A conditional law $K (s, \cdot)$ of a random element given $S$ composes with $S$ to give a regular conditional distribution given $σ (S)$ . Conditional law given a random element

[F2]

Integrating a product-measurable function against a probability kernel gives a measurable function of the kernel parameter. Measurability of integration against a kernel

[F3]

If $K (S, \cdot)$ is a supplied regular conditional distribution of $X$ given $σ (S)$ , then $\int f (x) K (S, d x)$ satisfies the conditional event-integral identities for nonnegative measurable $f$ ; for integrable real $f$ its signed integral is a conditional-expectation version. Conditional integration through a regular conditional law

[F4]

A conditional test's rejection probability given $S$ is a conditional-expectation version of $φ (X)$ . Conditional tests given a statistic

[F5]

The product sigma-algebra is generated by rectangles, so the second-coordinate projection is measurable; composing it with measurable $φ$ makes $(s, x) \mapsto φ (x)$ product-measurable. The product sigma-algebra and its finite iterates, Composition with a Borel measurable outer map preserves measurability

Proof

1.1F2F5given

The map $(s, x) \mapsto φ (x)$ is product-measurable by [F5]. Kernel integration [F2] therefore makes $q$ measurable. Since $K (s, \cdot)$ is a probability measure and $0 \leq φ \leq 1$ , every value satisfies $0 \leq q (s) \leq 1$ .

2.1A1F1F3F4step 1.1∎

Fix $η \in Θ_{0}$ . By [A1] and [F1], $K (S, \cdot)$ is a regular conditional distribution of $X$ given $σ (S)$ under $P_{η}$ . Applying [F3] to the bounded function $φ$ shows that for every $A \in σ (S)$ , $\int_{A} q (S) d P_{η} = \int_{A} φ (X) d P_{η}$ . Thus $q (S)$ is a version of the conditional rejection probability under this null law. The same kernel in [A1] defines the same $q$ for every $η$ , proving nuisance-freeness; only the individual laws' almost-sure qualifications are used.

Depends on

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Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §2.3, PDF pp. 33–35 (standard reference, not scraped)
Berkeley Stat 210A Course Reader, Testing with Nuisance Parameters, Conditional testing section (standard reference, not scraped)