CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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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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A measurable [0,1]-valued statistic need not be a valid p-value

Statement refuted

Every measurable statistic $p : X \to [0, 1]$ is a valid p-value for a given null hypothesis.

Facts & Assumptions

Given: A statistical experiment with a nonempty null hypothesis and a disjoint nonempty alternative, and the definition of a valid p-value.

[F1]

Statistical experiments: an experiment supplies a measurable data space and a probability law for each parameter.

[F2]

Valid p-values: a valid p-value must satisfy $P_{θ} (p (X) \leq u) \leq u$ for every null parameter and every $u \in [0, 1]$ .

[F3]

Test size and power: for the simple null $Θ_{0} = {θ_{0}}$ and a nonempty disjoint alternative, size is the rejection probability under $P_{θ_{0}}$ .

[F4]

A measurable function between measurable spaces: a statistic is measurable when all measurable target sets have measurable preimages.

[F5]

The expectation of an indicator is the probability of the event: the expectation of an event indicator is the event's probability.

[F6]

Every measurable rejection function defines a randomized test: every measurable function into $[0, 1]$ defines a binary randomized test.

Counterexample

1.1givenF1F4algebraconstruct

Let $X = {0, 1}$ with the full sigma-algebra, $Θ = {θ_{0}, θ_{1}}$ , $Θ_{0} = {θ_{0}}$ , and $Θ_{1} = {θ_{1}}$ . Set $P_{θ_{0}} ({0}) = P_{θ_{0}} ({1}) = 1 / 2$ and $P_{θ_{1}} ({0}) = 0$ , $P_{θ_{1}} ({1}) = 1$ . These masses define probability measures: they are nonnegative and sum to one, and on the two-point space countable additivity follows because at most two members of a disjoint family are nonempty. Define $p (0) = 0$ and $p (1) = 1$ . It is measurable because every subset of $X$ is measurable.

2.1F2F3F5F6step 1.1algebra∎

At $u = 1 / 4$ , ${x : p (x) \leq u} = {0}$ , so $P_{θ_{0}} (p (X) \leq 1 / 4) = 1 / 2 > 1 / 4$ . Thus $p$ is not valid by [F2]. Thresholding it at $1 / 4$ defines a test by [F6], rejects on ${0}$ , and has size $1 / 2$ by [F3,F5], so it is anti-conservative. The claimed implication is false.

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