Adaptive quantile estimation in deconvolution with unknown error distribution

TL;DR

This paper develops a minimax optimal, adaptive method for quantile estimation in deconvolution problems with unknown error distributions, filling a key gap in the statistical literature.

Contribution

It introduces a data-driven bandwidth selection for adaptive estimation and establishes optimal rates for distribution function estimation with unknown errors.

Findings

The proposed method achieves minimax optimality under natural conditions.

Adaptive estimation is successfully implemented with a data-driven bandwidth.

Application to real data demonstrates practical effectiveness.

Abstract

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.