M-Estimation in the partially linear model with Bernstein polynomials under shape constrains

ABSTRACT

We develope an M-estimator for partially linear models in which the nonparametric component is subject to various shape constraints. Bernstein polynomials are used to approximate the unknown nonparametric function, and shape constraints are imposed on the coefficients. Asymptotic normality of regression parameters and the optimal rate of convergence of the shape-restricted nonparametric function estimator are established under very mild conditions. Some simulation studies and a real data analysis are conducted to evaluate the finite sample performance of the proposed method.

Keywords:

• Bernstein polynomials
• Empirical process
• M-estimation
• Rate of convergence
• Shape constrains

Mathematics Subject Classification:

• 62G05
• 62G08

Acknowledgments

The authors are grateful to the Editor-in-Chief and the reviewers for constructive comments and helpful suggestions that lead to a great improvement of an earlier manuscript.

Funding

The research is supported by National Natural Science Foundation of China (11271039), Research Fund for the Doctoral Program of Higher Education of China and Fund for Collaborative Innovation Center on Capital Social Construction and Social Management.