Demystified: double robustness with nuisance parameters estimated at rate n-to-the-1/4

TL;DR

This paper explains the theoretical basis for double robustness with nuisance parameters estimated at a rate of n^(1/4), showing that under certain conditions, the variance of the estimator remains unaffected by nuisance parameter estimation.

Contribution

It provides a simple, rigorous explanation using the Middle Value Theorem for why nuisance parameters estimated at rate n^(1/4) or faster do not impact the variance of the main estimator.

Findings

Variance of the estimator is unaffected by nuisance parameter estimation at rate n^(1/4) or faster.

The sandwich estimator remains consistent even when nuisance parameters are estimated.

The explanation relies on basic smoothness conditions and Taylor expansion techniques.

Abstract

Have you also been wondering what is this thing with double robustness and nuisance parameters estimated at rate n^(1/4)? It turns out that to understand this phenomenon one just needs the Middle Value Theorem (or a Taylor expansion) and some smoothness conditions. This note explains why under some fairly simple conditions, as long as the nuisance parameter theta in R^k is estimated at rate n^(1/4) or faster, 1. the resulting variance of the estimator of the parameter of interest psi in R^d does not depend on how the nuisance parameter theta is estimated, and 2. the sandwich estimator of the variance of psi-hat ignoring estimation of theta is consistent.