Optimality of the quasi-score estimator in a mean-variance model with applications to measurement error models

We consider a regression of $y$ on $x$ given by a pair of mean and variance functions with a parameter vector $\theta$ to be estimated that also appears in the distribution of the regressor variable $x$. The estimation of $\theta$ is based on an extended quasi score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-$y$ unbiased estimating functions. Of special interest is the case where the distribution of $x$ depends only on a subvector $\alpha$ of $\theta$, which may be considered a nuisance parameter. In general, $\alpha$ must be estimated simultaneously together with the rest of $\theta$, but there are cases where $\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.We also study a number of special measurement error models in greater detail