Manuscript Psy03-B@1 · Psy03-B-v1.tex · proposition · 8ca3caf9

Proposition: Grid boundary certificate

Counterfactual sensitization and persistent vulnerability › Critical challenge without presumed monotonicity · lines 1526–1549 · proof included or linked

Psy03-B@1#prop:grid

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Suppose simultaneous bands $L_{j} \leq Q_{j} \leq U_{j}$ hold. Define

j_{L} = min {j : L_{j} < q_{*}}, j_{U} = min {j : U_{j} < q_{*}} .

Then

j_{L} \leq j^{*} \leq j_{U} .

Thus

j_{U, post} < j_{L, pre}

certifies a decrease of the grid critical-challenge index. No monotonicity of accuracy in bias is needed.

Proof. Every index below $j_{L}$ is certified nonfailing. The index $j_{U}$ , if finite, is certified failing.

Displayed formulas in this passage

unlabeled · line 1530
$j_{L} = min {j : L_{j} < q_{*}}, j_{U} = min {j : U_{j} < q_{*}} .$
unlabeled · line 1535
$j_{L} \leq j^{*} \leq j_{U} .$
unlabeled · line 1539
$j_{U, post} < j_{L, pre}$

Nearby in this section

definition: Definition: Grid critical challenge
section: Critical challenge without presumed monotonicity
proposition: Proposition: Conditional critical-root stability
section: Critical challenge without presumed monotonicity