Robust tests for equality of regression curves based on characteristic functions

Abstract

This paper focuses on the problem of testing the null hypothesis that the regression functions of several populations are equal under a general nonparametric homoscedastic regression model. It is well known that linear kernel regression estimators are sensitive to atypical responses. These distorted estimates will influence the test statistic constructed from them so the conclusions obtained when testing equality of several regression functions may also be affected. In recent years, the use of testing procedures based on empirical characteristic functions has shown good practical properties. For that reason, to provide more reliable inferences, we construct a test statistic that combines characteristic functions and residuals obtained from a robust smoother under the null hypothesis. The asymptotic distribution of the test statistic is studied under the null hypothesis and under root$-n$ contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed test with the classical one obtained using local averages. The reported numerical experiments show the advantage of the proposed methodology over the one based on Nadaraya–Watson estimators for finite samples. An illustration to a real data set is also provided and enables to investigate the sensitivity of the $p-$value to the bandwidth selection.

Keywords:

• Hypothesis testing
• nonparametric regression models
• robust estimation
• smoothing techniques

Acknowledgments

The authors wish to thank two anonymous referees and the Associate Editor for their valuable comments which led to an improved version of the original paper.

Disclosure statement

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

Additional information

Funding

This research was partially supported by the Universidad de Buenos Aires [grant number 20020170100022BA (G. Boente)] and Agencia Nacional de Promoción Científica y Tecnológica (ANPCYT) at Buenos Aires, Argentina [grant number PICT 2021-I-A-00260 (G. Boente)] and also by the Spanish Projects from the Ministry of Science and Innovation (MCIU/ AEI /10.13039/501100011033) [grant numbers PID2020-116587GB-I00 (G. Boente) and PID2020-118101GB-I00 (J. C. Pardo-Fernández)]. The research was begun while G. Boente was visiting the Universidade de Vigo supported by this last project.