Multiscale Methods for Shape Constraints in Deconvolution: Confidence Statements for Qualitative Features

TL;DR

This paper develops multiscale statistical methods to detect qualitative features like local monotonicity in deconvolution problems, providing theoretical guarantees and simulation evidence in moderately ill-posed settings.

Contribution

It introduces a novel multiscale testing framework for qualitative features in deconvolution, especially under polynomial decay error densities, with calibration based on Brownian motion modulus of continuity.

Findings

Multiscale tests effectively detect qualitative features in deconvolution.

The methods perform well both theoretically and in simulations.

Detection is feasible despite slow minimax rates for pointwise estimation.

Abstract

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.