Comparison of three estimators in Poisson errors-in-variables model with one covariate

A structural errors-in-variables model is investigated, where the
response variable follows a Poisson distribution. Assuming the
error variance to be known, we consider three consistent estimators
and compare their relative efficiencies by means of their
asymptotic covariance matrices. The comparison is made for
arbitrary error variances. The structural quasi-likelihood (QL)
estimator is based on a quasi score function, which is constructed
from a conditional mean-variance model. The corrected estimator is
based on an error-corrected likelihood score function. The
alternative estimator is constructed to remove the asymptotic bias
of the naive (i.e., ordinary maximum likelihood) estimator. It is
shown that the QL estimator is strictly more efficient than the
alternative estimator, and the latter one is strictly more
efficient than the corrected estimator.