= Efficient score in independent-error regression
{title2=$\widetilde\ell=(h-E_vh)\rho+(E_vh)\varepsilon/\tau^2$}
With unknown independent covariate and centered error distributions, take the regular <mean-preserving error tangent space>, let $h=\partial_\theta g_\theta$, $\rho=-f'/f$ and $\tau^2=E_f\varepsilon^2>0$. Under $E_f\rho=0$, $E_f(\varepsilon\rho)=1$ and finite second <moments>, <orthogonal projection> removes $(E_vh)(\rho-\varepsilon/\tau^2)$ from $h\rho$. Thus the <efficient score> and <efficient information> are $\widetilde\ell=(h-E_vh)\rho+(E_vh)\varepsilon/\tau^2$ and $\widetilde I=\operatorname{Var}_v(h)E_f\rho^2+(E_vh)^2/\tau^2$. The reduction to $h\varepsilon/\tau^2$ is valid for a <normal distribution> of errors or for constant $h$, but need not hold otherwise. Centered covariates and <logistic distribution> errors give the counterexample $\widetilde\ell=X\tanh(\varepsilon/2)$.
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