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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.

Write $e=y-g_{\theta }(x)$ and $h_{\theta }(x)=\partial g_{\theta }(x)/\partial \theta$. Holding the nuisance parameter $\eta$ fixed, the parametric score function is the derivative with respect to $\theta$ of the one-observation log-likelihood. The joint probability density function isBecause $\partial e/\partial \theta =-h_{\theta }(x)$, differentiating givesFor instance $E_{\eta }{h}_{\theta }(X{)}^2<\infty$ ensures a square-integrable score function. Its expected value is zero by independence and the centered normal distribution of the error. This is the Gaussian regression score.