Manuscript Oai282-A@1 · Oai282-A.tex · theorem · 5b6134ec

Theorem: Finite-confidence residual certificate

The finite local-Ward certificate › Optional bounded-score validation · lines 1125–1162 · proof included or linked

Oai282-A@1#thm:statistics

For $0 < α_{s} < 1$ , set

w_{k} = \frac{a _{k}}{n _{k}} i \sum Z_{k i}, ρ_{k} = a_{k} \frac{2}{n _{k}} lo g \frac{2 m}{α _{s}} + b_{k} .

Under Assumption ass:statistics,

Pr {∣ w_{k} - w_{k} ∣ \leq ρ_{k} for every k} \geq 1 - α_{s} - β .

Therefore, if a tolerance $ϵ_{k}$ is certified only when

∣ w_{k} ∣ + ρ_{k} \leq ϵ_{k} for every k,

the probability of certifying while some $∣ w_{k} ∣ > ϵ_{k}$ is at most $α_{s} + β$ .

Proof. Conditional Hoeffding inequalities give

Pr {\frac{1}{n _{k}} i \sum Z_{k i} - m_{k} > \frac{2}{n _{k}} lo g \frac{2 m}{α _{s}} Z} \leq \frac{α _{s}}{m} .

A union bound, integration over $Z$ , and intersection with $G$ prove simultaneous coverage. Every invalid acceptance lies outside that event.

Displayed formulas in this passage

unlabeled · line 1128
$w_{k} = \frac{a _{k}}{n _{k}} i \sum Z_{k i}, ρ_{k} = a_{k} \frac{2}{n _{k}} lo g \frac{2 m}{α _{s}} + b_{k} .$
unlabeled · line 1134
$Pr {∣ w_{k} - w_{k} ∣ \leq ρ_{k} for every k} \geq 1 - α_{s} - β .$
unlabeled · line 1140
$∣ w_{k} ∣ + ρ_{k} \leq ϵ_{k} for every k,$
unlabeled · line 1151
$Pr {\frac{1}{n _{k}} i \sum Z_{k i} - m_{k} > \frac{2}{n _{k}} lo g \frac{2 m}{α _{s}} Z} \leq \frac{α _{s}}{m} .$