Consistency of Robust Kernel Density Estimators

The kernel density estimator (KDE) based on a radial positive-semidefinite kernel may be viewed as a sample mean in a reproducing kernel Hilbert space. This mean can be viewed as the solution of a least squares problem in that space. Replacing the squared loss with a robust loss yields a robust kernel density estimator (RKDE). Previous work has shown that RKDEs are weighted kernel density estimators which have desirable robustness properties. In this paper we establish asymptotic L^1 consistency of the RKDE for a class of losses and show that the RKDE converges with the same rate on bandwidth required for the traditional KDE. We also present a novel proof of the consistency of the traditional KDE.