Abstract: Corrected score (Nakamura (1990); Stefanski (1989)) is an important consistent functional modeling method for covariate measurement error in nonlinear regression. Although its pathological behaviors are known to exacerbate with increasing error contamination, neither their nature nor severity is well understood. We conduct a detailed investigation with the loglinear model for count data in the presence of sizable measurement error. Our study reveals that multiple roots, estimate-finding failure, and skewness in distribution are common and may persist even when the sample size is large. These pathological behaviors are attributed to a surprising fact that the desirable trend of the corrected score always goes astray as the parameter space approaches extremes. A novel remedy is proposed to constrain the derivatives with additional estimating functions. The resulting trend-constrained corrected score may also substantially improve estimation efficiency. These findings and the estimation strategy shed light on the developments for other nonlinear models and for the nonparametric correction method.
Key words and phrases: Empirical likelihood, functional modeling, loglinear model, method of moments, multiple roots, nonlinear model, Poisson regression, random effects Poisson regression, trend-constrained corrected score.