Theorem: Complete-record compatibility and discrimination

Suppose the conditional fixed-shot contract holds for a true member $(j_{\ast },{\lambda }_{\ast })$, with

on $G$. Then

For two promised families, define the complete-record separation

If

the rule that reports the uniquely compatible family, and otherwise reports “inconclusive”, reports the true family with probability at least $1-\alpha -\beta$, uniformly on either promise class.

One conservative sufficient sample size is

Proof. For each coordinate indicator, conditional Hoeffding gives failure probability at most $2e^{-2n{c}_n^2}=\alpha /D$. A union bound yields simultaneous deviation at most $c_n$ from the average actual law. Total variation at most $e_{j_{\ast }}$ bounds every coordinate bias by $e_{j_{\ast }}$. This proves (eq:compatibility-coverage) after calibration failure is added.

On that simultaneous event, the true parameter is compatible. If a parameter of the other family were also compatible, the triangle inequality would give

contradicting (eq:separation-condition). Thus exactly the true family is compatible. The conservative sample bound gives $c_n\le (g-e_0-e_1)/4$, and therefore a strict separation.