Nonlinear regression models with general distortion measurement errors

Abstract

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.

Keywords:

• Multiplicative and additive distortion measurement errors
• local linear smoothing
• model checking
• restricted estimator

Mathematics Subject Classifications (2000):

• 62G05
• 62G08
• 62G20

Acknowledgments

The authors thank the editor, the associate editor, and the referee for their constructive suggestions that helped us to improve the early manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jun Zhang http://orcid.org/0000-0003-4332-5182

Additional information

Funding

Bingqing Lin's research was supported by the National Natural Science Foundation of China (NSFC) [grant number 11701386]. Gaorong Li's research is supported by the National Natural Science Foundation of China [grant numbers 11871001 and 11471029] and Beijing Natural Science Foundation [grant number 1182003].