Establish consistency of the fitted-PMM variance estimator

Establish consistency of the complete variance estimator for predictive mean matching when the matching models are fitted from the data, including the effects of estimated matching coefficients, conditional score means, and donor reuse.

Background

The paper develops RW-PMM, an approximate Robins–Wang variance estimator for multiple imputation by predictive mean matching. The approximation smooths the discontinuous donor-selection rule through differentiable donor probabilities, uses expected analysis scores to estimate the cross-moment, and aggregates analysis scores by recorded donor to account for dependence caused by donor reuse.

The appendix derives the cross-moment limit when matching coefficients are known and derives covariance expressions under shared donor values. However, the complete fitted-PMM estimator also involves estimated matching coefficients, potentially nonzero conditional means of donor-group scores, and imputation randomization. The paper explicitly identifies consistency of the full variance estimator under these fitted-model conditions as unresolved. Establishing this result would provide theoretical justification for the proposed RW-PMM procedure beyond the simulation evidence.