Quasi Score is more efficient than Corrected Score in a polynomial measurement error model

We consider a polynomial regression model, where the covariate is
measured with Gaussian errors. The measurement error variance is
supposed to be known. The covariate is normally distributed with
known mean and variance. Quasi Score (QS) and Corrected Score (CS)
are two consistent estimation methods, where the first makes use of
the distribution of the covariate (structural method), while the
latter does not (functional method). It may therefore be surmised
that the former method is (asymptotically) more efficient than the
latter one. This can, indeed, be proved for the regression
parameters. We do this by introducing a third, so-called Simple
Score (SS), estimator, the efficiency of which turns out to be
intermediate between QS and CS. When one includes structural and
functional estimators for the variance of the error in the
equation, SS is still more efficient than CS. When the mean and
variance of the covariate are not known and have to be estimated as
well, one can still maintain that QS is more efficient than SS for
the regression parameters.