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

摘要

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 preestimated. 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.