Asymptotic optimality of the quasi-score estimator in a class of linear score estimators

We prove that the quasi-score estimator in a mean-variance model is
optimal in the class of (unbiased) linear score estimators, in the
sense that the difference of the asymptotic covariance matrices of
the linear score and quasi-score estimator is positive
semi-definite. We also give conditions under which this difference
is zero or under which it is positive definite. This result can be
applied to measurement error models where it implies that the
quasi-score estimator is asymptotically more efficient than the
corrected score estimator.