Indirect Maximum Entropy Bandwidth

Abstract: This paper proposes a new method of bandwidth selection in kernel estimation of density and distribution functions motivated by the connection between maximisation of the entropy of probability integral transforms and maximum likelihood in classical parametric models. The proposed estimators are designed to indirectly maximise the entropy of the leave-one-out kernel estimates of a distribution function, which are the analogues of the parametric probability integral transforms. The estimators based on minimisation of the Cramer-von Mises discrepancy, near-solution of the moment-based estimating equations, and inversion of the Neyman smooth test statistic are discussed and their performance compared in a simulation study. The bandwidth minimising the Anderson-Darling statistic is found to perform reliably for a variety of distribution shapes and can be recommended in practice. The results will also be of interest to anyone analysing the cross-validation bandwidths based on leave-one-out estimates or evaluation of nonparametric density forecasts.