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

Jan Johannes, Bianca Neubert

TL;DR

The paper addresses nonparametric goodness-of-fit testing for the density of a strictly positive variable under multiplicative measurement error, leveraging Mellin transform techniques to convert multiplicative convolution into an estimable spectral form. It constructs a test statistic from a quadratic functional of Mellin transforms and establishes non-asymptotic radii in Mellin-Sobolev spaces, with data-driven adaptation via a Bonferroni-aggregated max-test. The authors derive quantile bounds for the test statistic, provide upper bounds on the testing radius across regularity classes, and analyze the adaptive cost (log factors) relative to non-adaptive procedures. Simulations illustrate finite-sample performance and demonstrate near-optimal adaptivity across ordinary and super-smooth settings, highlighting practical applicability to inverse problems such as multiplicative deconvolution in survival analysis.

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.

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

TL;DR

The paper addresses nonparametric goodness-of-fit testing for the density of a strictly positive variable under multiplicative measurement error, leveraging Mellin transform techniques to convert multiplicative convolution into an estimable spectral form. It constructs a test statistic from a quadratic functional of Mellin transforms and establishes non-asymptotic radii in Mellin-Sobolev spaces, with data-driven adaptation via a Bonferroni-aggregated max-test. The authors derive quantile bounds for the test statistic, provide upper bounds on the testing radius across regularity classes, and analyze the adaptive cost (log factors) relative to non-adaptive procedures. Simulations illustrate finite-sample performance and demonstrate near-optimal adaptivity across ordinary and super-smooth settings, highlighting practical applicability to inverse problems such as multiplicative deconvolution in survival analysis.

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.

Paper Structure

This paper contains 20 sections, 15 theorems, 159 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Mellin transform
  4. Multiplicative deconvolution
  5. Goodness-of-fit testing task
  6. Bounds for the quantiles of the test statistic
  7. Upper bound for the radius of testing
  8. Testing radius of a max-test
  9. Data-driven procedure via Bonferroni aggregation
  10. Testing radius of the max-test
  11. Naive choice of $\mathcal{K}$
  12. Optimized choice of $\mathcal{K}$
  13. Order of factor $\eta_{\mathcal{K}\vert n}^2$
  14. Simulation study
  15. Acknowlegdegments
  16. ...and 5 more sections

Key Result

Proposition 3.1

Let ass:well-definedness-testingass::error-well-definedeq::error-moments be satisfied and let $\alpha \in (0,1)$, $n\geq 2$ and $k\in{\mathds N}$. Consider the estimator ${\hat{q}}^2_{{{k}}}$ and the threshold $\tau_k(\alpha)$ defined in eq::estimatorq2kb and eq::quantiletau, respectively. Under the

Figures (2)

  • Figure 1: The boxplots represent the values of ${\hat{q}}^2_{{k}}$ for $f = f_1$ over 50 iterations. The horizontal lines indicate $q^2 = 0$. The triangles indicate different estimates for $0.1$-quantiles. The numbers given in the legend indicate the rejection rate of the corresponding test.
  • Figure 2: The boxplots represent the values of ${\hat{q}}^2_{{k}}$ for $f = f_2$ over 50 iterations. The horizontal lines indicate $q^2 = 0.5$. The triangles indicate different estimates for $0.1$-quantiles. The numbers given in the legend indicate the rejection rate of the corresponding test.

Theorems & Definitions (25)

  • Proposition 3.1: Bound for the quantiles of ${\hat{q}}^2_{{{k}}}$ under the null hypothesis
  • proof : Proof of \ref{['pr::boundsquantiles']}
  • Proposition 3.2: Bound for the quantiles of ${\hat{q}}^2_{{{k}}}$ under the alternative
  • proof : Proof of \ref{['pr::boundsquantiles-alternative']}
  • Proposition 4.1: Upper bound for the radius of testing of $\Delta_{k,\gamma/2}$
  • proof : Proof of \ref{['pr::upperbound-radius-testing']}
  • Corollary 1
  • Proposition 5.1: Bound for the quantiles of ${\hat{q}}^2_{{{k}}}$ under the null hypothesis
  • proof : Proof of \ref{['pr::boundsquantiles-updated']}
  • Proposition 5.2: Uniform radius of testing over regularity class $\mathcal{S}$
  • ...and 15 more