Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters

Abstract

We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x. The estimation of θ is based on an extended quasi score (QS) function. Of special interest is the case where the distribution of x depends only on a subvector α of θ, which may be considered a nuisance parameter. 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. Under unknown nuisance parameters we derive conditions under which the QS estimator is strictly more аfficient than the CS estimator. We focus on the loglinear Poisson, the Gamma, and the logit model.