Deconvolution with unknown error distribution

TL;DR

This paper develops nonparametric estimators for a density function from contaminated data with an unknown error distribution, using spectral methods and additional error samples, achieving minimax optimal rates under certain smoothness conditions.

Contribution

It introduces a spectral cutoff approach for deconvolution with unknown error distribution, providing convergence rates and demonstrating asymptotic optimality.

Findings

Estimators achieve minimax optimal convergence rates.

Method works with both known and unknown error densities.

Estimates are asymptotically optimal under Sobolev smoothness.

Abstract

We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density. We assume that an additional sample $\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\epsilon}$ and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density $f_{\epsilon}$, where it is assumed that $f_X$ satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density $f_X$ belongs to a Sobolev space $H_{\mathbh p}$ and $f_{\epsilon}$ is ordinary smooth or supersmooth.