Testing discontinuities in nonparametric regression

ABSTRACT

In nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13] and propose to further improve it. To achieve the goal, we first reveal that their method is less efficient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the residual variance and the total amount of discontinuities simultaneously. In both theory and simulation, we show that the proposed variance estimator has a smaller mean-squared error compared to the existing estimator, whereas the estimation efficiency for the total amount of discontinuities remains unchanged. Finally, we construct a new test procedure for detection of discontinuities using the proposed method; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings.

KEYWORDS:

• Asymptotic normality
• difference-based estimator
• jump point
• model selection
• nonparametric regression
• residual variance

Acknowledgments

The authors claim that the partial results in this paper follow the Chapter 5 of the first author's PhD dissertation at Hong Kong Baptist University [5], and they have not been published in any other scientific journals. The authors thank the editor, the associate editor and the referee for their constructive comments that led to a substantial improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Tiejun Tong's research was supported by the Hong Kong Baptist University grants FRG1/14-15/044, FRG2/15-16/019 and FRG2/15-16/038, and the National Natural Science Foundation of China grant (No. 11671338).