Maximum likelihood kernel density estimation

Methods for improving the basic kernel density estimator include variable locations, variable bandwidths (often called variable kernels) and variable weights. Currently these methods are implemented separately and via pilot estimation of variation functions derived from asymptotic considerations. In this paper, we propose a simple maximum likelihood procedure which allows (in its greatest generality) variation of all these quantities at once, bypasses asymptotics and explicit pilot estimation, and turns out to perform better. This maximum likelihood kernel density estimation, which involves a greatly overparametrised mixture model, works because the overall bandwidth (the geometric mean of individual bandwidths) is fixed. This overall bandwidth, in turn, is the single smoothing parameter of the methodology which has to be chosen separately. And the method has a further advantage: it automatically reduces, where appropriate, to a few-component mixture model which indicates and initialises parametric mixture modelling of the data. We set out simple algorithms, perform a substantial simulation study, give an illustrative example and provide some background theory. For computational and performance reasons we particularly recommend the variable location version of the methodology.