Validity of the categorical PCM under vector-valued nuisance conditions

Verify in detail that the asymptotic validity argument for the categorical projected covariance measure extends to the vector-valued residual and projection setting, including vector-form restatements and proofs of the supporting lemmas under the stated nuisance-estimation conditions.

Background

For categorical outcomes, the paper adapts the projected covariance measure by replacing its scalar projection direction and scalar residual product with a precision-weighted vector direction and an inner product of vector residuals. The authors state conditions intended to support the corresponding asymptotic validity argument.

The paper does not provide the detailed verification that the scalar PCM proof extends to this categorical vector formulation. Establishing that extension would provide the theoretical justification for the categorical PCM used in the empirical application, rather than relying primarily on simulation-based calibration.

References

The argument of \citet[Theorem~4]{lundborg2024projected} is therefore available with their product $\epsilon_i \xi_i$ replaced by $\langle \epsilon_i, \xi_i \rangle$, provided the conditions it places on the nuisance estimates are read in vector form. We record these conditions below; a detailed verification, including vector restatements of the supporting lemmas, is left to future work.

— Embedded Conditional Independence Tests for Large Language Model Generated Text with an Application to German Parliament Speeches  (2609.00946 - Simnacher et al., 1 Sep 2026) in Appendix, Section A.2, Subsection “Validity”