Cross-fitting rates under endogenous online logging

Establish general cross-fitting rates for nuisance estimation under endogenous adaptive online logging, beyond guarantees conditional on realized nuisance errors.

Background

The doubly robust coverage theorem is stated conditionally on the realized root-mean-square errors of the outcome-model and propensity estimators. This avoids imposing a universal statistical rate for nuisance estimation under adaptive data collection. However, the paper does not establish such a rate for arbitrary endogenous logging processes, where the observed data are selectively generated by a history-dependent policy and the calibration procedure itself may influence future actions.