The na\"ive estimator of a Poisson regression model with measurement errors

TL;DR

This paper analyzes the bias of the na"ive estimator in Poisson regression models with measurement errors, extending previous work to non-normal distributions and proposing a bias-corrected estimator with simulation validation.

Contribution

It generalizes the understanding of the na"ive estimator's bias beyond normal distributions and introduces a consistent bias-corrected estimator for broader applicability.

Findings

The na"ive estimator exhibits asymptotic bias under non-normal distributions.

The proposed estimator effectively reduces bias and improves estimation accuracy.

Simulation results demonstrate the estimator's robustness across different distributional assumptions.

Abstract

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.