Bernstein Polynomial Model for Nonparametric Multivariate Density

TL;DR

This paper introduces a Bernstein polynomial approach for nonparametric multivariate density estimation, demonstrating near-parametric convergence rates and superior performance over kernel methods through simulations and real data application.

Contribution

The paper develops a Bernstein polynomial model for multivariate density estimation, including an EM algorithm for maximum likelihood estimation and a change-point method for selecting model complexity.

Findings

Near-parametric convergence rate in mean χ²-divergence.

Outperforms kernel density estimates in finite samples.

Comparable to parametric methods in accuracy.

Abstract

In this paper, we study the Bernstein polynomial model for estimating the multivariate distribution functions and densities with bounded support. As a mixture model of multivariate beta distributions, the maximum (approximate) likelihood estimate can be obtained using EM algorithm. A change-point method of choosing optimal degrees of the proposed Bernstein polynomial model is presented. Under some conditions the optimal rate of convergence in the mean $\chi^2$-divergence of new density estimator is shown to be nearly parametric. The method is illustrated by an application to a real data set. Finite sample performance of the proposed method is also investigated by simulation study and is shown to be much better than the kernel density estimate but close to the parametric ones.