Maximum likelihood kernel density estimation: On the potential of convolution sieves

Methods for improving the basic kernel density estimator include variable locations, variable bandwidths and variable weights. Typically these methods are implemented separately and via pilot estimation of variation functions derived from asymptotic considerations. The starting point here is a simple maximum likelihood procedure which allows (in its greatest generality) variation of all these quantities at once, bypassing asymptotics and explicit pilot estimation. One special case of this approach is the density estimator associated with nonparametric maximum likelihood estimation (NPMLE) in a normal location mixture model. Another, closely associated with the NPMLE, is a kernel convolution sieve estimator proposed in 1982 but little used in practice to date. Simple algorithms are utilised, a simulation study is reported on, a method for bandwidth selection is investigated and an illustrative example is given. The simulations and other considerations suggest that the kernel convolution sieve provides an especially promising framework for further practical utilisation and development. 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.