Efficient and Robust Density Estimation Using Bernstein Type Polynomials

TL;DR

This paper introduces a Bernstein polynomial-based method for density estimation that is both efficient and robust, capable of approximating smooth distributions and estimating functionals like the mean, with proven consistency and practical effectiveness.

Contribution

The paper proposes a novel density estimation approach using Bernstein polynomials, including a simple method for selecting the polynomial degree, with theoretical guarantees and practical validation.

Findings

Achieves consistent maximum likelihood density estimates at near-parametric rates.

Provides a practical degree selection method for Bernstein polynomial models.

Demonstrates effectiveness on real data sets including microarray data.

Abstract

Method of parameterizing and smoothing the unknown underling distributions using Bernstein polynomials is proposed, verified and investigated. Any distribution with bounded and smooth enough density can be approximated by the proposed model. The approximating model turns out to be a mixture of beta distributions beta$(i+1, m-i+1)$, $i=0,\ldots, m$, for some optimal degree $m$. A simple change-point estimating method for choosing optimal degree $m$ of the Bernstein polynomials is also presented. The proposed methods give maximum likelihood density estimate which is consistent in $L_2$ distance at an almost parametric rate under some conditions. Simulation study shows that one can benefit from both the smoothness and the accuracy by using the proposed method. The proposed model can also be used to estimate some functional of the unknown distribution such as population mean. As illustration, the proposed methods are applied to three different type data sets including a microarray data set.