Consistent Kernel Density Estimation with Non-Vanishing Bandwidth
This addresses a foundational issue in nonparametric statistics for researchers and practitioners, offering a novel approach to bandwidth selection, though it is incremental in extending existing KDE methods.
The paper tackles the problem of achieving consistency in kernel density estimation with a fixed bandwidth, introducing the fixed-bandwidth KDE (fbKDE) and proving it consistently estimates continuous square-integrable densities, with experimental results showing it compares favorably to standard and variable bandwidth KDEs.
Consistency of the kernel density estimator requires that the kernel bandwidth tends to zero as the sample size grows. In this paper we investigate the question of whether consistency is possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, and prove that it consistently estimates any continuous square-integrable density. We also establish rates of convergence for the fbKDE with radial kernels and the box kernel under appropriate smoothness assumptions. Furthermore, in an experimental study we demonstrate that the fbKDE compares favorably to the standard KDE and the previously proposed variable bandwidth KDE.