Comparing Different Estimators in a Nonlinear Measurement Error Model

A nonlinear structural errors-in-variables model is investigated,
where the response variable has a density belonging to an
exponential family and the error-prone covariate follows a Gaussian
distribution. Assuming the error variance to be known, we consider
two consistent estimators in addition to the naive estimator. We
compare their relative efficiencies by means of their asymptotic
covariance matrices for small error variances. The structural quasi
score (SQS) estimator is based on a quasi score function, which is
constructed from a conditional mean-variance model. Consistency and
asymptotic normality of this estimator is proved. The corrected
score (CS) estimator is based on an error-corrected likelihood
score function. For small error variances the SQS and CS estimators
are approximately equally efficient. The polynomial model and the
Poisson regression model are explored in greater detail.