Manuscript Psy04-A@1 · Psy04-A-v1.tex · theorem · d5800e2a

Theorem: One-sided bounded classical-memory rejection

Finite-sample exclusion certificates › Fixed-sample rejection · lines 1228–1257 · proof included or linked

Psy04-A@1#thm:fixedtest, Psy04-A@1#eq:rejection

Suppose every actual classical null satisfies

E [S_{i} ∣ F_{i - 1}] \leq c, c = min {1, 3/4 + e_{C}},

at every attempted run. For a predeclared sample size $N$ and $0 < δ < 1$ , let

Q_{N} = \frac{1}{N} i = 1 \sum^{N} S_{i}, r_{N} (δ) = \frac{lo g ( 1/ δ )}{2 N} .

The rejection rule

Q_{N} - r_{N} (δ) > c

has Type-I error at most $δ$ , uniformly over that null.

Proof. Because $μ_{i} \leq c$ ,

i \sum (S_{i} - c) \leq i \sum (S_{i} - μ_{i}) .

Apply the upper-tail bound of Lemma lem:conditionalmgf.

Displayed formulas in this passage

Psy04-A@1#eq:rejection · line 1243
$Q_{N} - r_{N} (δ) > c$
unlabeled · line 1231
$E [S_{i} ∣ F_{i - 1}] \leq c, c = min {1, 3/4 + e_{C}},$
unlabeled · line 1237
$Q_{N} = \frac{1}{N} i = 1 \sum^{N} S_{i}, r_{N} (δ) = \frac{lo g ( 1/ δ )}{2 N} .$
unlabeled · line 1252
$i \sum (S_{i} - c) \leq i \sum (S_{i} - μ_{i}) .$