Robust improvement of efficiency using information on covariate distribution.

The marginal inference of an outcome variable can be improved by closely related covariates with a structured distribution. This differs from standard covariate adjustment in randomized trials, which exploits covariate-treatment independence rather than knowledge on the covariate distribution. Yet it can also be done robustly against misspecification of the outcome-covariate relationship. Starting with a standard estimating function involving only the outcome, we first use a working regression model to compute its conditional expectation given the covariates, and then remove the uninformative part under the covariate distribution model. This effectively projects the initial function onto the joint tangent space of the full data, thereby achieving local efficiency when the regression model is correct. Importantly, even with a faulty working model, the estimator remains unbiased as the subtracted term is always asymptotically centered. Further improvement is possible if the outcome distribution also has its own structure. To demonstrate the process, we consider three examples: one with fully parametric covariates, one with a covariate following a partial parametric model against others, and another with mutually independent covariates.