Manuscript Psy04-B@1 · Psy04-B-v1.tex · theorem · 62c2b6f0

Theorem: Fixed-sample exclusion certificate

Finite-sample exclusion conditional on the cut · lines 2109–2150 · proof included or linked

Psy04-B@1#thm:test, Psy04-B@1#eq:test

Under Assumption ass:history,

μ_{i} \leq c_{i} := min {1, 3/4 + e_{C, i}}

for every classical null. For predeclared $N$ and $0 < δ < 1$ , define

r_{N} (δ) = \frac{lo g ( 1/ δ )}{2 N} .

The rejection rule

\frac{1}{N} i = 1 \sum^{N} (S_{i} - c_{i}) > r_{N} (δ)

has Type-I error at most $δ$ , uniformly over the declared classical null.

For constant $e_{C}$ , it becomes

Q_{N} - r_{N} (δ) > min {1, 3/4 + e_{C}} .

Proof. Condition on the pre-run state and history. The fresh classical cut gives the bound $3/4$ ; the conditional complete-law discrepancy adds at most $e_{C, i}$ . Average any hidden pre-run state.

Since

i \sum (S_{i} - c_{i}) \leq i \sum (S_{i} - μ_{i}),

Lemma lem:concentration proves the rejection bound.

Displayed formulas in this passage

Psy04-B@1#eq:test · line 2122
$\frac{1}{N} i = 1 \sum^{N} (S_{i} - c_{i}) > r_{N} (δ)$
unlabeled · line 2112
$μ_{i} \leq c_{i} := min {1, 3/4 + e_{C, i}}$
unlabeled · line 2117
$r_{N} (δ) = \frac{lo g ( 1/ δ )}{2 N} .$
unlabeled · line 2132
$Q_{N} - r_{N} (δ) > min {1, 3/4 + e_{C}} .$
unlabeled · line 2146
$i \sum (S_{i} - c_{i}) \leq i \sum (S_{i} - μ_{i}),$

Theorem: Fixed-sample exclusion certificate

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