Accelerated Nonparametric Maximum Likelihood Density Deconvolution Using Bernstein Polynomial

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.