Previous studies on Poisson regression models with error-in-variables (EIV) assumed either a univariate EIV structure or multivariate EIV framework with all explanatory variables subject to error where the explanatory variable and error vectors are restricted to multivariate normal distributions. This study assumes that the explanatory variable and error vectors follow general distributions, with measurement error affecting only a subset of variables in the multivariate EIV framework. We define the partial-error naive estimator, derive its asymptotic bias and mean squared error, and propose a consistent estimator for the true parameter by correcting this bias. We also investigate a simplification of the new estimator when all components of the explanatory variable and error vectors are independent. This method is applicable even when the explanatory variable or error vectors follow a mixed distribution. Simulation studies and real data analysis are presented as illustrative examples to compare the performance of the partial-error naive estimator with that of the new estimator.

References

• Box, G. E. P. (1963). The effects of errors in the factor levels and experimental design. Technometrics, 5(2), 247–262. https://doi.org/10.1080/00401706.1963.10490078
• Burr, D. (1988). On errors-in-variables in binary regression-Berkson case. Journal of the American Statistical Association, 83(403), 739–743.
• Eshima, N., & Tabata, M. (2010). Entropy coefficient of determination for generalized linear models. Computational Statistics & Data Analysis, 54(5), 1381–1389. https://doi.org/10.1016/j.csda.2009.12.003
• Fuller, W. A. (1987). Measurement Error Models. Wiley.
• Geary, R. C. (1953). Non-linear functional relationship between two variables when one variable is controlled. Journal of the American Statistical Association, 48(261), 94–103. https://doi.org/10.1080/01621459.1953.10483458
• Guo, J. Q., & Li, T. (2002). Poisson regression models with errors-in-variables: Implication and treatment. Journal of Statistical Planning and Inference, 104(2), 391–401. https://doi.org/10.1016/S0378-3758(01)00250-6
• Huwang, L., & Huang, Y. H. (2000). On errors-in-variables in polynomial regression-Berkson case. Statistica Sinica, 10(3), 923–936.
• Jiang, F., & Ma, Y. (2022). Poisson regression with error corrupted high dimensional features. Statistica Sinica, 32(4), 2023–2046.
• Kenya National Bureau of Statistics (2019). FinAccess Survey 2019. https://knbs.or.ke(open in a new window)
• Kukush, A., & Schneeweiss, H. (2000). A comparison of asymptotic covariance matrices of adjusted least squares and structural least squares in error ridden polynomial regression. Sonderforschungsbereich, 386, 1–23.
• Kukush, A., Schneeweiss, H., & Wolf, R. (2004). Three estimators for the Poisson regression model with measurement errors. Statistical Papers, 45(3), 351–368. https://doi.org/10.1007/BF02777577
• Kukush, A., & Shklyar, S. (2002). Comparison of three estimators in Poisson errors-in-variables model with one covariate. Sonderforschungsbereich, 386, 1–23.
• Kurosawa, T., Hui, F. K. C., Welsh, A. H., Shinmura, K., & Eshima, N. (2020). On goodness-of-fit measures for Poisson regression models. Australian & New Zealand Journal of Statistics, 62(3), 340–366. https://doi.org/10.1111/anzs.v62.3
• McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in Econometrics (pp. 105–142). Academic Press.
• Nakamura, T. (1990). Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika, 77(1), 127–137. https://doi.org/10.1093/biomet/77.1.127
• Rosenthal, J. S. (2025). A First Look at Rigorous Probability Theory (2nd ed.). World Scientific Publishing.
• Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.
• 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, 94(2), 250–270. https://doi.org/10.1016/j.jmva.2004.05.002
• Shklyar, S., Schneeweiss, H., & Kukush, A. (2007). Quasi score is more efficient than corrected score in a polynomial measurement error model. Metrika, 65(3), 275–295. https://doi.org/10.1007/s00184-006-0076-5
• van der Vaart, A. W. (2012). Asymptotic Statistics. Cambridge University Press.
• Wada, K., & Kurosawa, T. (2023). The naive estimator of a Poisson regression model with a measurement error. Journal of Risk and Financial Management, 16(3), 16–30. https://doi.org/10.3390/jrfm16030186

To cite this article: Kentarou Wada & Takeshi Kurosawa (12 Dec 2025): Bias correction of partial-error in variables in a Poisson regression model, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2025.2588849

To link to this article: https://doi.org/10.1080/24754269.2025.2588849