Robust Kernel Density Estimation by Scaling and Projection in Hilbert Space
This work addresses the issue of sample contamination in datasets for researchers and practitioners using kernel density estimation, though it is incremental as it builds on existing KDE methods.
The paper tackles the problem of robust nonparametric density estimation by introducing a scaled and projected kernel density estimator (SPKDE) that addresses sample contamination, demonstrating through numerical experiments and consistency results that it asymptotically recovers the uncontaminated density under certain conditions.
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.