Discontinuities in robust nonparametric regression with α-mixing dependence

ABSTRACT

The main idea of the paper is to introduce a robust regression estimation method under an α-mixing dependence assumption, staying free of any parametric model restrictions while also allowing for some sudden changes in the unknown regression function. The sudden changes in the model may correspond to discontinuity points (jumps) or higher order breaks (jumps in corresponding derivatives) as well. We firstly derive some important statistical properties for local polynomial M-smoother estimates and we will propose a statistical test to decide whether some given point of interest is significantly important for a change to occur or not. As the asymptotic distribution of the test statistic depends on quantities which are left unknown we also introduce a bootstrap algorithm which can be used to mimic the target distribution of interest. All necessary proofs are provided together with some experimental results from a simulation study and a real data example.

KEYWORDS:

• Local polynomial M-smoothers
• robust nonparametric regression
• discontinuity in regression
• change points
• α-mixing dependence
• moving blocks bootstrap

AMS SUBJECT CLASSIFICATION:

• 62G08
• 62G10
• 62G20
• 62G35

Acknowledgments

The authors would like to express thanks to the editor and both referees for reading the paper thoroughly and raising useful comments and suggestions in order to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The smoothing parameter in nonparametric regression is well known for being crucial for the final performance. Therefore, we only considered the optimal value of the parameter given by the corresponding robust version of the Cross-Validation criterion. For smaller values of the bandwidth parameter the model overfits the data and it becomes unlikely to reveal true change-points. On the other side, for large bandwidth parameter the final model oversmooth the data and artificial change-points are likely to occur in order to account for the variability which is not taken into account by the model.

2. In addition to the simulation results presented in this paper we also considered a comparison between homoscedastic and heteroscedastic scenario. However, due to the fact that we are only interested in the estimation at three distant points from the domain of interest, there is no visible difference between homoscedastic and heteroscedastic approach. Moreover, the bootstrap algorithm used for the statistical test is defined using the model-based approach and thus, the estimation of the scale function is not an issue as well. We also considered other forms of dependence for the error terms: for instance, AR(1) or GARCH(1,1) such that the sequence is strictly stationary.

Additional information

Funding

This work was partially supported by the grant 15-09663S of Hušková. The research of Maciak was partially supported by the grant P402/12/G097.