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

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A conditionally superuniform stage-2 p-value preserves conditional error

Statement

Under the hypotheses and notation of Conditional-error budget for two-stage adaptation, let $p_{2}$ be a measurable stage-2 p-value with values in $[0, 1]$ . Choose an $F_{1}$ -measurable representative $A$ of the original conditional-error class that takes values in $[0, 1]$ everywhere; such a representative exists by clipping any version on its null exceptional set. Changing to this version changes $R_{2}$ only on a null set. Suppose that for every fixed $u \in [0, 1]$ , $E_{0} [1_{{p_{2} \leq u}} ∣ F_{1}] \leq u almost surely .$ Then the adapted rejection event $R_{2} = {p_{2} \leq A}$ satisfies $E_{0} [1_{R_{2}} ∣ F_{1}] \leq A almost surely,$ and hence $P_{0} (R_{2}) \leq α$ . The conditional p-value inequality is assumed separately for each fixed $u$ ; no common exceptional set over all $u \in [0, 1]$ is required.

Facts & Assumptions

Given: AC; one fixed null law $P_{0}$ ; a sub-sigma-algebra $F_{1}$ ; an original rejection event with a conditional-error version $A$ chosen to take values in $[0, 1]$ everywhere, and $E_{0} A = P_{0} (R_{0}) \leq α$ ; and a measurable stage-2 p-value $p_{2}$ satisfying the displayed conditional inequality for every fixed $u \in [0, 1]$ .

[A1]

AC supplies the conditional-expectation existence interface used for $A$ and the rejection-event conditional expectation. The Axiom of Choice

[F1]

Conditional-expectation classes have measurable integrable versions satisfying the event-integral identities and are unique almost surely. Conditional expectation as an ae class

[F2]

A p-value statistic is measurable and takes values in $[0, 1]$ . Valid p-values

[F3]

If $Z$ is bounded and $F_{1}$ -measurable, then $E_{0} [Z X ∣ F_{1}] = Z E_{0} [X ∣ F_{1}]$ . Taking out what is known

[F4]

Conditional expectation preserves increasing limits of nonnegative variables and decreasing limits of integrable variables. Conditional monotone convergence

[F5]

Conditional expectation is linear, preserves constants, and preserves order. Basic algebra and order properties of conditional expectation

[F6]

If the adapted conditional rejection probability is at most $A$ almost surely and $E_{0} A \leq α$ , then the adapted null rejection probability is at most $α$ . Conditional-error budget for two-stage adaptation

Proof

technique · approximate the interim-measurable threshold from above by finite-valued thresholds, then pass to the decreasing limit

1.1A1F1F2F3F5givenchoosealgebra

Conditional order and the constant rule [F5] give $0 \leq A \leq 1$ almost surely. Clipping a version on the measurable null set where it leaves $[0, 1]$ preserves its conditional-expectation class and changes $R_{2}$ only on a null set, so take $0 \leq A \leq 1$ everywhere. For each integer $n \geq 1$ , define $A_{n} = 2^{- n} ⌈ 2^{n} A ⌉$ . Its level sets are ${A_{n} = 0} = {A = 0}$ and ${A_{n} = k 2^{- n}} = {(k - 1) 2^{- n} < A \leq k 2^{- n}}$ for $1 \leq k \leq 2^{n}$ , so they are in $F_{1}$ . The inequality $⌈ 2 x ⌉ \leq 2 ⌈ x ⌉$ shows $A_{n + 1} \leq A_{n}$ , and $0 \leq A_{n} - A < 2^{- n}$ , hence $A_{n}$ is finite-valued and $A_{n} ↓ A$ . Write its values as $t_{n, k} = k 2^{- n}$ , $k = 0, \dots, 2^{n}$ , and put $S_{n, k} = {A_{n} = t_{n, k}}$ . By [F2], the events ${p_{2} \leq t_{n, k}}$ are measurable. Let $Q_{n} = E_{0} [1_{{p_{2} \leq A_{n}}} ∣ F_{1}]$ . The disjoint partition by $S_{n, k}$ , linearity [F5], and taking out the bounded known indicator [F3] give $Q_{n} = \sum_{k = 0}^{2^{n}} 1_{S_{n, k}} E_{0} [1_{{p_{2} \leq t_{n, k}}} ∣ F_{1}] \leq \sum_{k = 0}^{2^{n}} t_{n, k} 1_{S_{n, k}} = A_{n}$ almost surely. For all $n$ these inequalities hold on one probability-one set, because only countably many fixed dyadic thresholds occur.

2.1F4F5step 1.1algebra

The events $B_{n} = {p_{2} \leq A_{n}}$ decrease to $B = {p_{2} \leq A}$ . Their complement indicators are bounded and hence integrable, and increase to $1_{B^{c}}$ . Thus [F4] gives $E_{0} [1_{B_{n}^{c}} ∣ F_{1}] ↑ E_{0} [1_{B^{c}} ∣ F_{1}]$ almost surely. By linearity and the constant rule [F5], this is equivalent to $Q_{n} ↓ Q : = E_{0} [1_{B} ∣ F_{1}]$ . Taking limits in step 1.1 and using $A_{n} ↓ A$ yields $Q \leq A$ almost surely.

3.1F6step 2.1∎

Apply [F6] to $R_{2} = B$ : its conditional rejection probability is the $Q$ of step 2.1 and is at most $A$ , while the original conditional-error budget has expectation at most $α$ . Therefore $P_{0} (R_{2}) \leq α$ .

Source notes

The dyadic argument handles the random threshold without assuming a regular conditional distribution or a jointly chosen version of all fixed-threshold conditional probabilities. Conditional superuniformity at each fixed threshold is sufficient because the approximating thresholds use only countably many dyadic values.

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)