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

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. 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. 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 efficient than the $CS$ estimator. We focus on the loglinear Poisson, the Gamma, and the logit model.