Density estimation and nonparametric inferences using maximum likelihood weighted kernels

Abstract

We show that maximum likelihood weighted kernel density estimation offers a unified approach to density estimation and nonparametric inferences. For density estimation, the approach is a generalisation of the standard kernel density estimator that allows the weights attached to each kernel to be chosen by maximum likelihood, instead of being set to n −1 from the outset (see also Jones, M.C., and Henderson, D.A. (2005), ‘Maximum Likelihood Kernel Density Estimation’, Technical Report 01/05, Department of Statistics, The Open University, UK). For nonparametric inferences, the approach offers a natural, smoothed analogue to empirical likelihood (Owen, A.B. (2001), Empirical Likelihood, Boca Raton, FL: Chapman and Hall/CRC) for inferences on functionals of the underlying distribution, such as its mean or median. Numerical results demonstrate that the proposed method is comparable to the standard kernel density estimator (of the same bandwidth) for density estimation, but can offer noticeable small-sample improvements over empirical likelihood for inferences when the underlying distribution is continuous.