Goodness-of-fit testing from observations with multiplicative measurement error

TL;DR

This paper develops a nonparametric goodness-of-fit test for positive variables with multiplicative measurement error, using Mellin transform-based estimators and multiple testing procedures, with theoretical guarantees and simulations.

Contribution

It introduces a new non-asymptotic testing method based on Mellin transform estimators, providing adaptive procedures with controlled error rates in the presence of multiplicative errors.

Findings

Derived non-asymptotic testing radii and rates for Mellin-Sobolev spaces.

Proposed data-driven testing procedures with performance close to non-adaptive tests.

Validated the methods through simulation studies with various densities.

Abstract

Given observations from a positive random variable contaminated by multiplicative measurement error, we consider a nonparametric goodness-of-fit testing task for its unknown density in a non-asymptotic framework. We propose a testing procedure based on estimating a quadratic functional of the Mellin transform of the unknown density and the null. We derive non-asymptotic testing radii and testing rates over Mellin-Sobolev spaces, which naturally characterize regularity and ill-posedness in this model. By employing a multiple testing procedure with Bonferroni correction, we obtain data-driven procedures and analyze their performance. Compared with the non-adaptive tests, their testing radii deteriorate by at most a logarithmic factor. We illustrate the testing procedures with a simulation study using various choices of densities.