# Accelerated Nonparametric Maximum Likelihood Density Deconvolution Using Bernstein Polynomial

Research paper by Zhong Guan

Indexed on: 24 Jan '16Published on: 24 Jan '16Published in: Statistics - Methodology

#### Abstract

A new method for deconvoluting density in measurement error models using the Bernstein type polynomial model which is actually a finite mixture of specific beta distributions is proposed and studied. The change-point detection method is used to choose an optimal model degree. Based a contaminated sample of size $n$, the 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 $f$ has continuous $2k$-th derivative with $k>1$. In addition if the error distribution is generalized normal with not too high noise level the mean chi-squared distance between the proposed density estimate and $f$ is shown to be possibly $\mathcal{O}(n^{-1}\log^2 n)$.