Multivariate EDF tests for uniformity, normality,spherical and elliptical symetry, and independence based on a Brownian sheet deconstruction

2026-05-17

AI Summary

This study addresses goodness-of-fit testing for multivariate distributions, focusing on uniformity, normality, spherical and elliptical symmetry, and independence. It introduces a novel approach based on the decomposition of Brownian sheets. Under the null hypothesis, the problem is transformed via probability integral transforms into testing uniformity over the unit hypercube, and interactions among coordinates are effectively disentangled using a zero-margin measure decomposition. The method uniquely integrates the Gaussian process decomposition of Brownian sheets with empirical distribution functions, substantially enhancing sensitivity to joint dependence structures. Simulation studies demonstrate that the proposed test achieves power comparable to or exceeding that of current state-of-the-art methods, particularly excelling in detecting complex dependency patterns.

Abstract

This paper extends a recently proposed family of EDF-based goodness-of-fit procedures for the hypercube $[0,1]^p$ - the m-test and the s-test - which are based on a unique deconstruction of the $p$-parameter Brownian sheet into independent Gaussian processes. We use the fact that whenever a null hypothesis implies a joint distribution that factorizes into independent continuous components after a suitable mapping, the problem can be reduced to a uniformity test on the hypercube via componentwise probability integral transforms. Specifically, we introduce and analyze new procedures derived from these principles for testing uniformity on the hypersphere $S^p$, as well as multivariate normality, spherical and elliptical symmetry, and independence in $R^p$. The methodology is based on the decomposition of finite signed measures into zero-marginal components to isolate coordinate interactions. Empirical power comparisons show that these extended procedures are highly competitive with existing methods in the statistical literature, demonstrating particular sensitivity to coordinate-based dependencies and joint dependency structures.

Problem

• uniformity
• normality
• spherical symmetry
• elliptical symmetry
• independence

Innovation

• Brownian sheet decomposition
• EDF-based goodness-of-fit
• multivariate uniformity test
• coordinate interaction
• probability integral transform