On consistency for wavelet estimator of regression function based on biased samples under extended negatively dependence

ABSTRACT

For a nonparametric regression model with biased samples under extended negatively dependence, we concern with the estimation problem of the regression function over Sobolev space. On the basis of the wavelet kernel, we propose a wavelet estimator and investigate its consistency properties under mild conditions. For the proposed wavelet estimator, we establish the pth mean consistency based on moment inequality and get the complete consistency by virtue of exponential inequality and truncation technique. It should be pointed out that the complete consistency of wavelet estimator is a novel theoretical result in the nonparametric regression model with biased samples. In addition, the effectiveness of the proposed wavelet estimator is demonstrated on some numerical experiments.

KEYWORDS:

• Nonparametric regression model
• wavelet estimator
• biased sample
• extended negatively dependence
• complete consistency

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62G05
• 62G08
• 62G20

Acknowledgments

The authors would like to thank the reviewers for their helpful comments that led to a much improved version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper is supported by the China Postdoctoral Science Foundation (Grant no. 2022M710867).