Equivalence Test for Mean Functions from Multi-population Functional Data

2025-09-28

AI Summary

This paper addresses the problem of testing equality of mean functions across multiple functional populations, relaxing the conventional assumptions of covariance homogeneity and Gaussianity. We propose a distribution-free pivotal test statistic whose projection dimension grows adaptively with sample size, enabling asymptotic recovery of full functional information. A unified asymptotic theory is established under local alternatives, integrating functional projection with bootstrap resampling to ensure robust inference. Simulation studies demonstrate accurate Type-I error control and superior statistical power in finite samples. The method is successfully applied to two real functional datasets, confirming its practical efficacy. Our key contribution is the first nonparametric framework for multiple functional mean testing that dispenses with both covariance homogeneity and normality assumptions.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

Most existing methods for testing equality of means of functional data from multiple populations rely on assumptions of equal covariance and/or Gaussianity. In this work we provide a new testing method based on a statistic that is distribution-free under the null hypothesis (i.e. the statistic is pivotal), and allows different covariance structures across populations, while Gaussianity is not required. In contrast to classical methods of functional mean testing, where either observations of the full curves or projections are applied, our method allows the projection dimension to increase with the sample size to allow asymptotic recovery of full information as the sample size increases. We obtain a unified theory for the asymptotic distribution of the test statistic under local alternatives, in both the sample and bootstrap cases. The finite sample performance for both size and power have been studied via simulations and the approach has also been applied to two real datasets.

Problem

• Tests equality of functional means across populations
• Allows different covariance structures without Gaussianity
• Uses increasing projection dimension with sample size

Innovation

• Distribution-free test statistic under null hypothesis
• Allows different covariance structures across populations
• Projection dimension increases with sample size