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

We consider a regression of yy on xx given by a pair of mean and variance functions with a parameter vector θθ to be estimated that also appears in the distribution of the regressor variable xx. The estimation of θθ 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-yy unbiased estimating functions. Of special interest is the case where the distribution of xx depends only on a subvector αα of θθ, which may be considered a nuisance parameter. In general, αα must be estimated simultaneously together with the rest of θθ, but there are cases where αα 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.