Semiparametric estimation with learned nuisance functions

Control nuisance bias and data reuse to complete an inference argument

Return to the eligible-population feature mean \mu=E[Y] with some outcomes missing. Under missing at random and positivity, the observed-data functional uses an outcome regression and an observation probability. These are nuisance functions: learning them is part of estimating the target, not a replacement for defining it.

The guide builds one complete argument for this mean. An augmented contribution cancels first-order population sensitivity to nuisance error, leaving a product of learning errors. A concrete local-regression calculation explains how prediction accuracy can meet that requirement. Cross-fitting separates training from evaluation so the remaining empirical error can be controlled fold by fold. The final walkthrough connects those results to an estimator and a justified standard error.

What you’ll learn

• Control the effect of learning: derive the exact nuisance-bias remainder, assess sufficient learning rates, and distinguish orthogonality from cross-fitting.
• Construct and justify an estimator: derive the observed-data influence function, implement held-out contributions, establish uncertainty, and diagnose identification, variance, or remainder failures.

The completion endpoint is SP-13: reproduce the missing-outcome mean argument from target and assumptions through estimator, proof, and interpretation. SP-10 through SP-12 supply its learning-rate and empirical-term steps; SP-14 checks failure diagnosis. A covariance or regression target requires its own derivative and remainder rather than automatic reuse of the formula.