An approximation of the corrected naive estimator for a Poisson regression model with a measurement error

We provide an approximation of the corrected naive estimator of unknown parameters in a Poisson regression model along with the measurement error.The corrected naive estimator is a consistent estimator of the true parameter.However, the corrected naive estimator does not always have an explicit expression when the explanatory variable and measurement error are general distributions.This study proposes an approximate corrected naive estimator as a simple closed form and an approximation of the corrected naive estimator for general distributions.The proposed estimator can be applied if the corrected naive estimator does not have an explicit expression.Furthermore, the asymptotic bias of the approximate corrected naive estimator is close to that of the corrected naive estimator.Additionally, we show that the asymptotic bias of the new estimator is smaller than that of the Naive estimator for sufficiently small error variance.As illustrative examples, we provide simulation studies and real data analyses that compare the performances of the naive, corrected naive, and approximate corrected naive estimators.