Robust Kernel Density Estimation by Scaling and Projection in Hilbert Space

Robert A. Vandermeulen; Clayton D. Scott

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arXiv · 1411.4378

Robust Kernel Density Estimation by Scaling and Projection in Hilbert Space

Robert A. Vandermeulen·Clayton D. Scott

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Abstract

While robust parameter estimation has been well studied in parametric density estimation, there has been little investigation into robust density estimation in the nonparametric setting. We present a robust version of the popular kernel density estimator (KDE). As with other estimators, a robust version of the KDE is useful since sample contamination is a common issue with datasets. What "robustness" means for a nonparametric density estimate is not straightforward and is a topic we explore in this paper. To construct a robust KDE we scale the traditional KDE and project it to its nearest weighted KDE in the $L^2$ norm. This yields a scaled and projected KDE (SPKDE). Because the squared $L^2$ norm penalizes point-wise errors superlinearly this causes the weighted KDE to allocate more weight to high density regions. We demonstrate the robustness of the SPKDE with numerical experiments and a consistency result which shows that asymptotically the SPKDE recovers the uncontaminated density under sufficient conditions on the contamination.

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BibTeX RIS

Robert A. Vandermeulen, Clayton D. Scott. 2014-11-17. Robust Kernel Density Estimation by Scaling and Projection in Hilbert Space. https://arxiv.org/abs/1411.4378

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