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

TL;DR

This paper develops new EDF-based goodness-of-fit tests for multivariate distributions, leveraging a Brownian sheet decomposition, and demonstrates their effectiveness in detecting various distributional properties and dependencies.

Contribution

It introduces novel procedures for testing uniformity, normality, symmetry, and independence in multivariate data using a Brownian sheet deconstruction approach.

Findings

Procedures are highly competitive with existing methods.

Enhanced sensitivity to coordinate dependencies.

Effective in detecting joint distributional properties.

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.