Robust estimators for additive models using backfitting

ABSTRACT

Additive models provide an attractive setup to estimate regression functions in a nonparametric context. They provide a flexible and interpretable model, where each regression function depends only on a single explanatory variable and can be estimated at an optimal univariate rate. Most estimation procedures for these models are highly sensitive to the presence of even a small proportion of outliers in the data. In this paper, we show that a relatively simple robust version of the backfitting algorithm (consisting of using robust local polynomial smoothers) corresponds to the solution of a well-defined optimisation problem. This formulation allows us to find mild conditions to show Fisher consistency and to study the convergence of the algorithm. Our numerical experiments show that the resulting estimators have good robustness and efficiency properties. We illustrate the use of these estimators on a real data set where the robust fit reveals the presence of influential outliers.

KEYWORDS:

• Fisher-consistency
• kernel weights
• robust estimation
• smoothing

AMS SUBJECT CLASSIFICATION:

• MSC 62G35
• 62G08

Supplemental material

The supplementary file includes two Sections labelled S.1 and S.2

[S.1: Empirical influence] To study the sensitivity of the robust backfitting with respect to single outliers, we provide a numerical study of the empirical influence function.

[S.2: Real data example] This section contains partial residuals plots for each explanatory variable, both for the classical and robust estimators. For the classical estimators, partial residuals are given for the complete data set and for the data without the outliers.

Acknowledgements

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

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by Grants pip 112-201101-00339 from Consejo Nacional de Investigaciones Científicas y Técnicas, PICT 2014-0351 from Fondo para la Investigación Científica y Tecnológica, anpcyt and 20020130100279BA from the Universidad de Buenos Aires at Buenos Aires, Argentina (G. Boente and A. Martínez) and Discovery Grant 250048-11 of the Natural Sciences and Engineering Research Council of Canada (M. Salibián Barrera). This research was begun while Alejandra Martínez was visiting the University of British Columbia supported by the Emerging Leaders in the Americas Program from the Canadian Bureau for International Education and Foreign Affairs and International Trade Canada.