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

Proposition: Prospective power

Finite-sample exclusion conditional on the cut · lines 2165–2198 · proof included or linked

Psy04-B@1#prop:power

If an alternative has conditional means

μ_{i} \geq c + g, g > 0,

with constant null allowance $c$ , then the fixed-sample rule has failure-to-reject probability at most

exp [- 2 N (g - r_{N} (δ))^{2}]

when $r_{N} (δ) < g$ .

A sufficient sample size for Type-I error at most $δ$ and failure-to-reject probability at most $β$ is

N \geq \frac{( lo g ( 1/ δ ) + lo g ( 1/ β ) ) ^{2}}{2 g ^{2}} .

Proof. Failure to reject implies

i \sum (S_{i} - μ_{i}) \leq N (r_{N} (δ) - g) .

Use the lower tail in Lemma lem:concentration. The sample-size condition ensures $g \geq r_{N} (δ) + r_{N} (β)$ .

Displayed formulas in this passage

unlabeled · line 2168
$μ_{i} \geq c + g, g > 0,$
unlabeled · line 2173
$exp [- 2 N (g - r_{N} (δ))^{2}]$
unlabeled · line 2181
$N \geq \frac{( lo g ( 1/ δ ) + lo g ( 1/ β ) ) ^{2}}{2 g ^{2}} .$
unlabeled · line 2192
$i \sum (S_{i} - μ_{i}) \leq N (r_{N} (δ) - g) .$