On deconvolution of distribution functions

TL;DR

This paper investigates the nonparametric estimation of a distribution function from noisy observations, developing optimal and adaptive estimators with practical algorithms and demonstrating their effectiveness through numerical experiments.

Contribution

It introduces rate optimal and adaptive estimators for distribution functions under measurement errors, with explicit convex optimization formulations and practical implementation.

Findings

Developed minimax optimal estimators based on empirical characteristic functions

Proposed explicit convex optimization-based affine estimators

Numerical results show good practical performance of the estimators

Abstract

The subject of this paper is the problem of nonparametric estimation of a continuous distribution function from observations with measurement errors. We study minimax complexity of this problem when unknown distribution has a density belonging to the Sobolev class, and the error density is ordinary smooth. We develop rate optimal estimators based on direct inversion of empirical characteristic function. We also derive minimax affine estimators of the distribution function which are given by an explicit convex optimization problem. Adaptive versions of these estimators are proposed, and some numerical results demonstrating good practical behavior of the developed procedures are presented.