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The Naive Estimator of a Poisson Regression Model with a Measurement Error

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  • Kentarou Wada

    (Department of Applied Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 1628601, Japan)

  • Takeshi Kurosawa

    (Department of Applied Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 1628601, Japan)

Abstract

We generalize the naive estimator of a Poisson regression model with a measurement error as discussed in Kukush et al. in 2004. 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 naive estimator and derive its asymptotic bias and asymptotic mean squared error (MSE). The requirements for the existence of the naive estimator can be expressed using an implicit function, which the requirements can be deduced by the characteristic of the Poisson regression models. In addition, using the implicit function obtained from the system of equations of the Poisson regression models, we propose a consistent estimator of the true parameter by correcting the bias of the naive estimator. As illustrative examples, we present simulation studies that compare the performance of the naive estimator and new estimator for a Gamma explanatory variable with a normal error or a Gamma error.

Suggested Citation

  • Kentarou Wada & Takeshi Kurosawa, 2023. "The Naive Estimator of a Poisson Regression Model with a Measurement Error," JRFM, MDPI, vol. 16(3), pages 1-15, March.
    • Handle: RePEc:gam:jjrfmx:v:16:y:2023:i:3:p:186-:d:1092285
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    References listed on IDEAS

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    1. Shklyar, S. & Schneeweiss, H., 2005. "A comparison of asymptotic covariance matrices of three consistent estimators in the Poisson regression model with measurement errors," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 250-270, June.
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