Determine whether the apparent high-dimensional remainder-term difficulty is an artifact of the von Mises decomposition

Determine whether the apparent failure to establish a sufficiently small remainder term for the standard G-computation or augmented inverse probability weighting estimator under proportional asymptotics is merely an artifact of separately decomposing the von Mises expansion into empirical-process and remainder terms, such that the combined terms may nevertheless be shown to be small.

Background

The paper analyzes G-computation estimators based on canonical generalized linear models when the number of covariates is large relative to the sample size. Under sample splitting, standard theory controls the estimator through empirical-process terms and second-order remainder terms, but the authors explain that the required convergence of prediction errors may fail when the covariate-to-sample-size ratio does not vanish.

The appendix examines whether this theoretical obstacle reflects a genuine failure of the estimator or instead results from analyzing components of the von Mises expansion separately. The issue is important because the combined empirical-process and prediction-error contributions could potentially cancel or remain small even when the individual remainder-term conditions are not satisfied. The paper does not resolve this question; it proceeds with a more detailed decomposition and concludes that the usual influence function omits a term that cannot be well approximated.