Adaptive minimax testing for circular convolution

TL;DR

This paper develops adaptive minimax goodness-of-fit tests for circular convolution models with measurement error, achieving near-optimal performance across various regularity classes in a non-asymptotic setting.

Contribution

It introduces adaptive testing procedures that perform optimally over unknown regularity and ill-posedness parameters, with a detailed analysis of their performance and limitations.

Findings

Adaptive tests achieve near-optimal rates with a logarithmic penalty.

Lower bounds show the unavoidable nature of the log-factor loss.

Performance is illustrated for Sobolev spaces and different error densities.

Abstract

Given observations from a circular random variable contaminated by an additive measurement error, we consider the problem of minimax optimal goodness-of-fit testing in a non-asymptotic framework. We propose direct and indirect testing procedures using a projection approach. The structure of the optimal tests depends on regularity and ill-posedness parameters of the model, which are unknown in practice. Therefore, adaptive testing strategies that perform optimally over a wide range of regularity and ill-posedness classes simultaneously are investigated. Considering a multiple testing procedure, we obtain adaptive i.e. assumption-free procedures and analyse their performance. Compared with the non-adaptive tests, their radii of testing face a deterioration by a log-factor. We show that for testing of uniformity this loss is unavoidable by providing a lower bound. The results are illustrated considering Sobolev spaces and ordinary or super smooth error densities.