  We generalize the na\"ive estimator of a Poisson regression model with
measurement errors as discussed in Kukush et al. [1]. The explanatory variable
is not always normally distributed as they assume. In this study, we assume
that the explanatory variable and measurement error are not limited to a normal
distribution. We clarify the requirements for the existence of the na\"ive
estimator and derive its asymptotic bias and asymptotic mean squared error
(MSE). In addition, we propose a consistent estimator of the true parameter by
correcting the bias of the na\"ive estimator. As illustrative examples, we
present simulation studies that compare the performance of the na\"ive
estimator and new estimator for a Gamma explanatory variable with a normal
error or a Gamma error.
