 Solution

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.