LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Selected-size conditional validity yields a conditionally superuniform p-value

Statement

Assume AC. Let $(Ω, F, P_{0})$ be a probability space, let $F_{1} \subseteq F$ be the interim sigma-algebra, let $N$ be an $F_{1}$ -measurable sample size taking values in a nonempty set $M \subseteq N_{> 0}$ , and let $p_{N}$ be the selected stage-2 p-value. For $m \in M$ , put $D_{m} = {N = m}$ . Suppose that for every $m \in M$ , every fixed $u \in [0, 1]$ , and every $H \in F_{1}$ , $P_{0} (H \cap D_{m} \cap {p_{N} \leq u}) \leq u P_{0} (H \cap D_{m}) .$ Then for every fixed $u \in [0, 1]$ , $E_{0} [1_{{p_{N} \leq u}} ∣ F_{1}] \leq u almost surely .$ Thus conditional validity within each selected-size stratum implies conditional superuniformity after an interim-measurable finite sample size is selected, even when the set of allowed sample sizes is countably infinite.

Facts & Assumptions

Given: AC; a probability space $(Ω, F, P_{0})$ ; a sub-sigma-algebra $F_{1}$ ; a nonempty set $M \subseteq N_{> 0}$ ; an $F_{1}$ -measurable map $N : Ω \to M$ ; a measurable p-value $p_{N} : Ω \to [0, 1]$ ; and the displayed event inequality for every $m, u, H$ .

[A1]

AC is assumed for the library's conditional-expectation existence interface. The Axiom of Choice

[F1]

Under AC, $E_{0} [X ∣ F_{1}]$ is the unique almost-sure class of integrable $F_{1}$ -measurable versions satisfying the event-integral identities. Conditional expectation as an ae class

[F2]

A p-value is a measurable statistic with values in $[0, 1]$ ; its null validity is the superuniform event inequality for each fixed $u$ . Valid p-values

[F3]

A probability measure is a countably additive measure of total mass one. Probability measures and probability spaces Measures on sigma-algebras

[F4]

The integral of an event indicator is its probability. The expectation of an indicator is the probability of the event

[F5]

Conditional expectation preserves nonnegativity and order. Basic algebra and order properties of conditional expectation

[F6]

The nonnegative integral is monotone: if $f \leq g$ , then $\int f d μ \leq \int g d μ$ . Monotonicity and nonnegative homogeneity of the nonnegative integral

Proof

technique · partition the event by selected sample size, then use the defining event integrals to verify the conditional bound

1.1givenF2F3

Fix $u \in [0, 1]$ and $H \in F_{1}$ . The event ${p_{N} \leq u}$ is measurable by the p-value statistic definition [F2]. The events $D_{m}$ , $m \in M$ , are pairwise disjoint and partition $Ω$ . Countable additivity [F3] gives $P_{0} (H \cap {p_{N} \leq u}) = \sum_{m \in M} P_{0} (H \cap D_{m} \cap {p_{N} \leq u})$ and $P_{0} (H) = \sum_{m \in M} P_{0} (H \cap D_{m})$ . Summing the assumed stratum inequalities over the increasing enumeration of $M$ (equivalently, taking finite partial sums and then their limits) yields $P_{0} (H \cap {p_{N} \leq u}) \leq u P_{0} (H)$ .

2.1A1F1F4F5F6step 1.1choosealgebra∎

Let $Y_{u}$ be a version of $E_{0} [1_{{p_{N} \leq u}} ∣ F_{1}]$ , which exists by [A1] and [F1]. The event identities in [F1] and [F4] turn step 1.1 into $\int_{H} Y_{u} d P_{0} \leq u P_{0} (H)$ for every $H \in F_{1}$ . Also $Y_{u} \geq 0$ almost surely by [F5]. If $P_{0} (Y_{u} > u) > 0$ , then for some integer $k \geq 1$ the set $H_{k} = {Y_{u} \geq u + 1 / k} \in F_{1}$ has positive probability, because ${Y_{u} > u} = ⋃_{k \geq 1} H_{k}$ . Monotonicity [F6] gives $\int_{H_{k}} Y_{u} d P_{0} \geq (u + 1 / k) P_{0} (H_{k}) > u P_{0} (H_{k})$ , contradicting the preceding event-integral bound. Thus $Y_{u} \leq u$ almost surely. The argument is for each fixed $u$ and does not assert a common exceptional set over uncountably many thresholds.

Source notes

The lemma is the local countable-stratification argument that makes the selected-size conditional law explicit. Sugiura–Uesaka describe fresh nonoverlapping stage samples and warn that the selected stage-2 sample size cannot simply replace the prespecified value in inverse-normal weights; Minas states conditional invariance for the stage-2 p-value after adaptation. Neither citation replaces the event-partition proof above.

Depends on

Used by

Calibrated selected sample size and an uncalibrated weight change Example

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Sources

Kenkichi Sugiura and Hiroyuki Uesaka, Theoretical Basis of Adaptive Designs for Clinical Trials in Phase II and III, Japanese Journal of Biometrics 29, Special Issue 1 (2008), S33–S52 (standard reference, not scraped)
Giorgos Minas, Multivariate Global Testing and Adaptive Designs, PhD thesis, University of Warwick (2013) (standard reference, not scraped)