Accelerated Nonparametric Maximum Likelihood Density Deconvolution Using Bernstein Polynomial

TL;DR

This paper introduces a new maximum likelihood density deconvolution method using Bernstein polynomial models, achieving optimal convergence rates and outperforming kernel estimators in simulations.

Contribution

It proposes a novel Bernstein polynomial-based maximum likelihood approach for density deconvolution with known error distribution, including an optimal model selection technique.

Findings

Achieves optimal convergence rate of $k^{-1}n^{-1+1/k} ext{log}^3 n$ under certain smoothness conditions.

Outperforms deconvolution kernel density estimators in small sample simulations.

Demonstrates practical utility through real data application.

Abstract

A new maximum likelihood method for deconvoluting a continuous density with a positive lower bound on a known compact support in additive measurement error models with known error distribution using the approximate Bernstein type polynomial model, a finite mixture of specific beta distributions, is proposed. The change-point detection method is used to choose an optimal model degree. Based on a contaminated sample of size $n$, under an assumption which is satisfied, among others, by the generalized normal error distribution, the optimal rate of convergence of the mean integrated squared error is proved to be $k^{-1}\mathcal{O}(n^{-1+1/k}\log^3 n)$ if the underlying unknown density has continuous $2k$th derivative with $k>1$. Simulation shows that small sample performance of our estimator is better than the deconvolution kernel density estimator. The proposed method is illustrated by a real data application.