Corrected Score with Sizable Covariate Measurement Error: Pathology and Remedy.

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. In this article, we conduct a detailed investigation with the log-linear 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 they may persist even when the sample size is practically large. Furthermore, these pathological behaviors are attributed to a surprising fact that 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 the estimation efficiency. These findings and estimation strategy shed light on the developments for other nonlinear models as well, including logistic and Cox regression models, and for nonparametric correction.