Choose (or blend) the combination from the conditioning of Sigma

Given a permutation covariance Sigma of the representations, returns the screen's decision. Under "threshold", if the condition number of the correlation matrix exceeds threshold, choose the Cauchy (which inverts nothing), else the quadratic. Under "shrink" (the default), shrink the correlation matrix toward the identity, R_a = (1 - a) R + a I, choosing the intensity a so the condition number drops to threshold; a = 0 is the full quadratic and a = 1 is the diagonal covariance (which ignores the cross-representation dependence, as the Cauchy does). The shrunk covariance is returned for the quadratic to use.

Usage

riposte_screen(Sigma, control = riposte_screen_control())

Arguments

Sigma

a permutation covariance of the representations (the closed-form covariance without adjustment, the pooled empirical covariance with it).

control

a riposte_screen_control() list.

Value

a list with method, the condition number of the correlation matrix, and either choice ("quadratic"/"cauchy", for "threshold") or the shrinkage lambda, the shrunk Sigma, and the achieved condition_shrunk (for "shrink").

Details

Choosing the combination from the condition number does not disturb the test's exact level, but the reason differs by path. Without covariance adjustment, Sigma is the closed-form permutation covariance — a function of the outcomes (fixed under the sharp null) and the design, NOT of the realized assignment — so the condition number is constant across the randomization reference, an ancillary quantity. With covariance adjustment, Sigma is the pooled empirical covariance of the residual statistics, which depends on the realized draws; it is no longer ancillary, but it is a symmetric function of the exchangeable pooled set, which is the property exactness actually needs (see dev/theory-adjusted-exactness.md).

The shrinkage-intensity rule here is a sensible default, not the settled Project 1 result; expect it to change. Set lambda in riposte_screen_control() to fix the intensity yourself. Exact level holds for any shrinkage rule; the rule affects power only.