Multivariate locally adaptive density estimation

Abstract

Multivariate versions of variable bandwidth kernel density estimators can lead to improvement over kernel density estimators using global bandwidth choices. These estimators are more flexible and better able to model complex (multimodal) densities. In this work, two variable bandwidth estimators are discussed: the balloon estimator which varies the smoothing matrix with each estimation point and the sample point estimator which uses a different smoothing matrix for each data point. A binned version of the sample point estimator is developed that, for various situations in low to moderate dimensions, exhibits less error (MISE) than the fixed bandwidth estimator and the balloon estimator. A practical implementation of the sample point estimator is shown through simulation and example to do a better job at reconstructing features of the underlying density than fixed bandwidth estimators. Computational details, including parameterization of the smoothing matrix, are discussed throughout.