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

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.