Adaptivity to an unknown covariate distribution ($E[\dot{\ell }\mid X]=0$)

If the parametric score function has zero conditional expectation given a covariate $X$, it is orthogonal to the centered functions of $X$ that form the covariate-density nuisance tangent space. Its efficient score then equals its parametric score function, and the unknown covariate distribution causes no loss of Fisher information. This applies to Gaussian regression scores and more generally to regular conditional models with unrestricted covariate distribution and no additional nuisance components.