Reference
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.