Goodness-of-fit for distributions on metric spaces

2026-09-29

AI Summary

This study addresses goodness-of-fit testing for distributions on separable metric spaces, particularly the asymptotic theoretical challenges arising under composite hypotheses. It proposes a general framework based on distance profiles that integrates empirical process theory with Bahadur-type expansions to accommodate composite settings. The authors derive the asymptotic distributions of the test statistics and establish their consistency. Furthermore, they prove the conditional asymptotic validity of the multiplier bootstrap under both simple and composite null hypotheses. This framework unifies goodness-of-fit testing paradigms on metric spaces and overcomes key theoretical bottlenecks in composite hypothesis inference. Simulation studies and real-world applications, including spherical data analysis, confirm its superiority and practical utility.

Abstract

We propose a general goodness-of-fit framework for distributions on separable metric spaces. Under suitable identifiability conditions, probability distributions are characterized by distance profiles, which motivates their use in goodness-of-fit testing, for simple and composite null hypotheses. For composite null hypotheses, parameter estimation is incorporated via a Bahadur-type expansion, and the asymptotic distribution of the empirical process for distance profiles is obtained under the null. We define test statistics based on this empirical process and derive their asymptotic null distributions. We further study the behavior of the proposed tests under fixed and local alternatives, establishing consistency results. Multiplier bootstrap procedures are developed, and their conditional asymptotic validity is established under both simple and composite null hypotheses. The methodology is illustrated with simulation studies and real-data applications for data on the sphere, hyperboloid, and simplex.

Problem

• goodness-of-fit
• metric spaces
• distance profiles
• composite null hypotheses
• non-Euclidean data

Innovation

• Goodness-of-fit
• Metric spaces
• Distance profiles
• Multiplier bootstrap
• Bahadur expansion