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

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Bonferroni simultaneous coverage

Statement

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments) with nonempty parameter set $Θ$ . Fix $m \geq 1$ coordinate targets $ψ_{i} : Θ \to Ψ_{i}$ and procedures $C_{i} : X \to P (Ψ_{i})$ , $i \in [m]$ , where $P (Ψ_{i})$ denotes the set of all subsets of $Ψ_{i}$ . For every $θ \in Θ$ , suppose each membership event $A_{i} (θ) : = {x \in X : ψ_{i} (θ) \in C_{i} (x)}$ belongs to $A$ and has marginal coverage $P_{θ} (A_{i} (θ)) \geq 1 - α_{i},$ where $0 \leq α_{i} \leq 1$ and $\sum_{i = 1}^{m} α_{i} \leq α \leq 1$ . Then, for every $θ \in Θ$ , $P_{θ} {ψ_{i} (θ) \in C_{i} (X) for every i \in [m]} \geq 1 - α .$ Thus the family of marginal confidence sets has simultaneous coverage at least $1 - α$ . No independence assumption is needed.

Facts & Assumptions

Given: A statistical experiment with nonempty parameter set, a fixed finite family of targets and procedures, measurable membership events $A_{i} (θ)$ for every $i$ and $θ$ , and marginal coverage bounds $P_{θ} (A_{i} (θ)) \geq 1 - α_{i}$ with $\sum_{i} α_{i} \leq α \leq 1$ .

[F1]

Statistical experiments: for each $θ$ , $P_{θ}$ is a probability measure on the sample sigma-algebra.

[F2]

Confidence sets and coverage: a confidence-set procedure has a measurable membership section for every candidate parameter, and its coverage at $θ$ is the probability that the reported set contains $θ$ .

[F3]

Basic identities for a probability measure: complements satisfy $P (A^{c}) = 1 - P (A)$ , and a finite union of measurable events has probability at most the sum of their probabilities.

[A1]

For every $θ \in Θ$ and $i \in [m]$ , $A_{i} (θ) \in A$ and $P_{θ} (A_{i} (θ)) \geq 1 - α_{i}$ .

Proof

technique · direct

Given: The experiment with nonempty parameter set, targets, confidence procedures, and marginal coverage assumptions in the statement.

1.1givenA1F2F3construct

For each $i$ , define the induced full-parameter confidence set $D_{i} (x) : = {η \in Θ : ψ_{i} (η) \in C_{i} (x)}$ . Each membership section of $D_{i}$ is $A_{i} (η)$ , measurable by [A1], so [F2] applies and its coverage at $θ$ is $P_{θ} (A_{i} (θ)) \geq 1 - α_{i}$ . The miss event $B_{i} (θ) : = A_{i} (θ)^{c}$ is measurable and, by [F3], satisfies $P_{θ} (B_{i} (θ)) = 1 - P_{θ} (A_{i} (θ)) \leq α_{i}$ .

2.1givenF1F3step 1.1algebra∎

Failure of simultaneous coverage is $⋃_{i = 1}^{m} B_{i} (θ)$ . By finite subadditivity [F3] and step 1.1, its probability is at most $\sum_{i = 1}^{m} α_{i} \leq α$ . Taking the complement gives the stated simultaneous coverage bound. This argument allows the membership events to be dependent.

Depends on

Used by

Simultaneous coverage for dependent interval sets Example

Dependency tree · two levels

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Sources

William Fithian, STAT 210A Lecture 26: Multiple Testing, PDF pp. 9–11 (standard reference, not scraped)
Jelle J. Goeman and Aldo Solari, The Sequential Rejection Principle of Familywise Error Control, Section 3, Eq. (13) (standard reference, not scraped)