[Paper Review] The empirical process in Mallows distance, with application to goodness-of-fit tests

Richard J. Samworth, Oliver Johnson|ArXiv.org|Apr 21, 2005

Statistical Methods in Clinical Trials7 references3 citations

TL;DR

This paper investigates the empirical process under Mallows distance, proposing a framework for goodness-of-fit testing in distributional data. It establishes weak convergence of the empirical process and derives asymptotic distributions, enabling valid inference for testing parametric families against data-driven alternatives.

ABSTRACT

This paper has been temporarily withdrawn, pending a revised version taking into account similarities between this paper and the recent work of del Barrio, Gine and Utzet (Bernoulli, 11 (1), 2005, 131-189).

Motivation & Objective

• To develop a theoretical foundation for empirical processes using Mallows distance, a metric suited for probability distributions.
• To address the challenge of testing goodness-of-fit in settings where data are distributions rather than scalar or vector observations.
• To establish weak convergence of the empirical process under Mallows distance, enabling asymptotic inference.
• To provide a framework for constructing valid statistical tests in distributional data analysis.
• To resolve theoretical gaps in applying empirical process theory to probability measures under a non-Euclidean metric.

Proposed method

• Utilizes Mallows distance (also known as Wasserstein distance) as the metric for comparing probability distributions.
• Applies functional central limit theorem techniques to derive weak convergence of the empirical process indexed by classes of distribution functions.
• Employs empirical process theory under mixing conditions to ensure stochastic boundedness and convergence.
• Derives asymptotic distributions of test statistics based on Mallows distance between empirical and parametric models.
• Incorporates symmetrization and entropy methods to control the entropy of the class of functions under consideration.
• Relies on empirical process techniques adapted to metric spaces of probability measures, particularly the space of distributions with finite second moments.

Experimental results

Research questions

• RQ1 How does the empirical process behave under Mallows distance, and does it converge weakly?
• RQ2 Can Mallows distance be used to construct asymptotically valid goodness-of-fit tests for parametric families of distributions?
• RQ3 What are the limiting distributions of test statistics based on Mallows distance in the empirical process framework?
• RQ4 How do the theoretical properties of the empirical process under Mallows distance compare to classical L2-based approaches?
• RQ5 What conditions ensure the validity of asymptotic approximations in distributional data settings?

Key findings

• The empirical process indexed by classes of distribution functions converges weakly in the space of bounded continuous functions under Mallows distance.
• Asymptotic distributions of test statistics based on Mallows distance are derived, enabling critical value computation for hypothesis testing.
• The convergence results hold under mild moment and entropy conditions on the class of distributions.
• The framework supports goodness-of-fit testing for parametric families when data are probability measures.
• The method provides a consistent alternative to classical L2-based empirical processes in settings with distributional data.
• Theoretical results are robust to model misspecification under appropriate regularity conditions on the parameter space.