A note on the relation between one--step, outcome regression and IPW--type estimators of parameters with the mixed bias property

Abstract: Bruns-Smith et al. (2025) established an algebraic identity between the one-step estimator and a specific outcome regression-type estimator for a class of parameters that forms a strict subset of the class introduced in Chernozhukov et al. (2022), assuming both nuisance functions are estimated as linear combinations of given features. They conjectured that this identity extends to the broader mixed bias class introduced in Rotnitzky et al. (2021). In this note, we prove their conjecture and further extend the result to allow one of the nuisance estimators to be non-linear. We also relate these findings to the work of Robins et al. (2007), who established other identities linking one-step estimators to outcome regression-type and IPW-type estimators.