On multivariate associated kernels for smoothing general density functions

TL;DR

This paper introduces multivariate associated kernel estimators with correlation structures for density estimation, demonstrating improved performance over diagonal kernels, especially for complex, multimodal distributions, through theoretical properties and simulation studies.

Contribution

It extends existing associated kernel estimators by incorporating correlation structures and analyzes their properties and performance, especially for bivariate beta kernels.

Findings

Correlation-structured kernels outperform diagonal kernels in simulations.

Full bandwidth matrices improve density estimation for multimodal distributions.

Good performance demonstrated on real election data.

Abstract

Multivariate associated kernel estimators, which depend on both target point and bandwidth matrix, are appropriate for partially or totally bounded distributions and generalize the classical ones as Gaussian. Previous studies on multivariate associated kernels have been restricted to product of univariate associated kernels, also considered having diagonal bandwidth matrices. However, it is shown in classical cases that for certain forms of target density such as multimodal, the use of full bandwidth matrices offers the potential for significantly improved density estimation. In this paper, general associated kernel estimators with correlation structure are introduced. Properties of these estimators are presented; in particular, the boundary bias is investigated. Then, the generalized bivariate beta kernels are handled with more details. The associated kernel with a correlation structure is built with a variant of the mode-dispersion method and two families of bandwidth matrices are discussed under the criterion of cross-validation. Several simulation studies are done. In the particular situation of bivariate beta kernels, it is therefore pointed out the very good performance of associated kernel estimators with correlation structure compared to the diagonal case. Finally, an illustration on real dataset of paired rates in a framework of political elections is presented.