Three Estimators for the Poisson Regression Model with Measurement Errors

We consider two consistent estimators for the parameters of the
linear predictor in the Poisson regression model, where the
covariate is measured with errors. The measurement errors are
assumed to be normally distributed with known error variance
sigma_u^2. The SQS estimator, based on a conditional mean-variance
model, takes the distribution of the latent covariate into account,
and this is here assumed to be a normal distribution. The CS
estimator, based on a corrected score function, does not use the
distribution of the latent covariate. Nevertheless, for small
sigma_u^2, both estimators have identical asymptotic covariance
matrices up to the order of sigma_u^2. We also compare the
consistent estimators to the naive estimator, which is based on
replacing the latent covariate with its (erroneously) measured
counterpart. The naive estimator is biased, but has a smaller
covariance matrix than the consistent estimators (at least up to
the order of sigma_u^2.)