Multivariate Uniformity Tests

• Multivariate uniformity tests are statistical procedures that assess whether multidimensional samples are uniformly distributed across domains by leveraging geometric transformations and probabilistic theories.
• They employ techniques like empirical process decompositions, optimal transport based ranks, and graph-based methods to detect deviations, dependence, and clustering in data.
• These tests are crucial for applications in simulation validation, spatial statistics, and high-dimensional data analysis, offering consistent detection of non-uniformity under various alternatives.

Multivariate uniformity tests comprise a collection of statistical methodologies for assessing whether a random sample in $\mathbb{R}^d$, or on a manifold such as the hypersphere or hypercube, arises from a uniform distribution. Unlike the univariate case, absence of a natural ordering in multidimensional spaces necessitates sophisticated approaches that address geometry, dependence structure, and computational complexity. The field integrates probabilistic theory (e.g., empirical process limits, optimal transport, Brownian sheets), multivariate rank concepts, and functionals of geometric graphs or depth functions. Such tests are critical in simulation validation, spatial statistics, high-dimensional data analysis, and the paper of directional datasets.

1. Fundamental Principles and Test Construction

The definition of multivariate uniformity typically hinges on whether the sample is absolutely continuous (with respect to Lebesgue or Hausdorff measure) and supported on a known domain, such as $[0,1]^d$ or the $d$-dimensional sphere $S^{d-1}$. The diversity of test designs reflects (a) the absence of a canonical ordering, (b) the rich geometry of possible supports, and (c) targeted sensitivity to types of alternatives such as dependence, clustering, or multimodality.

Key principles for constructing uniformity tests include:

• Probability Integral Transformation (PIT): Extends univariate uniformity concepts to the multivariate setting by transforming observations, often via coordinatewise or more general transformations, to the unit hypercube or hypersphere. In the case of dependent/coherent structures, transformation may require learning a normalizing flow or similar mapping (Shtembari et al., 2022).
• Reduction to Univariate Problems: Many methods seek to "collapse" the multidimensional question to one or several univariate tests, either by projection, composition of coordinate depths, or construction of summary random variables whose univariate distribution reflects the uniformity hypothesis (Klebanov et al., 2018).

2. Empirical Process Approaches: Brownian Sheets and Extensions

The empirical process for a $p$-variate uniform sample on $C = [0,1]^p$ admits a functional limit in the form of the multiparameter Brownian sheet. Decomposition techniques, such as those detailed for the $p$-Brownian sheet (Cabaña et al., 7 Sep 2025), stratify the process into $2^p$ independent components ("ramps" or "tents") corresponding to faces of the hypercube. Each component, $T_H$, admits an explicit Karhunen–Loève series with the squared $L^2$ norm $\|T_H\|^2$ having a distribution given by infinite sums of weighted, independent $\chi^2$ random variables.

This enables test construction as follows:

• Calculate the empirical process $W_n(t)$ based on indicator functions of multivariate order.
• Decompose $W_n$ into independent components and compute $\|T_{n,H}\|^2$ for each $H \subset \{1,\dots,p\}, H\ne\varnothing$.
• Use the limiting distribution $P_H(x) = P(\|T_H\|^2 \le x)$ to obtain one $p$-value per component.
• Combine $p$-values via the minimum ("$m$-test") or sum ("$s$-test"), with the latter converging to a $\chi^2_{2^p - 1}$-distribution under the null (Cabaña et al., 7 Sep 2025).

These tests are consistent: under any alternative, at least one component norm diverges, ensuring detection of non-uniformity.

3. Optimal Transport, Multivariate Ranks, and Characteristic Functions

Another general methodology uses the optimal transport framework to define "multivariate ranks" by mapping original data and a synthetic sample from the target (uniform) distribution onto a fixed reference grid, typically via the Monge problem:

$$\hat G_n = \argmin_{G:\mathcal{Z}_N \to \mathcal{G}_N\text{ bijective}} \sum_{i=1}^N \|G(Z_i) - Z_i\|^2$$

(Hlávka et al., 31 Jul 2025). Here, $\mathcal{Z}_N$ is the pooled sample, and $\mathcal{G}_N$ is a deterministic grid approximating the uniform measure.

A test statistic compares the empirical characteristic functions $\hat\phi_n(t),\ \hat\phi_m^{(0)}(t)$ of the respective multivariate ranks for the observed and null data:

$$D_{n,m} = \frac{nm}{n+m} \int_{\mathbb{R}^p} |\hat\phi_n(t) - \hat\phi_m^{(0)}(t)|^2 w(t)\, dt$$

Critical values in the simple null case may be precomputed via Monte Carlo owing to the distribution-free property of the ranks. The approach extends to composite hypotheses by incorporating bootstrap calibration to correct for parameter estimation effects (Hlávka et al., 31 Jul 2025).

Notably, analogous optimal-transport-based tests yield multivariate extensions of classical two-sample rank tests, preserving exact distribution-free properties in the simple null case and supporting explicit comparison of efficiency bounds (e.g., Pitman efficiency, Chernoff-Savage type results) (Deb et al., 2021, Shi et al., 2021).

4. Graph-Based, Geometric, and Random Spacings Tests

Tests based on geometric graphs generalize the concept of spacings to higher dimensions. A principal example is the maximal spacing test, which considers the largest volume of a fixed shape (e.g., a ball or simplex) that can fit into the observation window without containing any sample point: $$A_n = \sup \{ r\ :\ x + rA \subset K\setminus \{X_1,\ldots,X_n\}\},\qquad V_n = |A_n| = r^d$$ Under uniformity, $nV_n - \log n -(d-1)\log\log n - B$ converges in law to the Gumbel distribution (Henze, 2017). Consistency follows from coupling arguments that link the process under alternatives to the uniform case.

Related graph-based functionals, particularly those based on the construction of random geometric graphs with edges below a certain threshold, yield statistics based on sums of edge lengths raised to a power $\beta$:

$$L_n(\beta) = \frac{1}{2} \sum_{x, y \in X_n, x \ne y} \mathbb{I}(\|x-y\| < r_n)\|x-y\|^\beta$$

(Ebner et al., 2018). Standardizations and variance computations enable construction of test statistics whose limiting distributions are central or noncentral $\chi^2$ under the null and contiguous alternatives, respectively.

5. Uniformity Tests on the Hypersphere, Sobolev and Projection-Based Methods

Testing uniformity for data on the $(q+1)$-dimensional unit hypersphere $S^q$ is addressed via several classes of tests:

• Sobolev tests utilize spectral decompositions on the hypersphere, expanding test statistics in terms of eigenfunctions (Gegenbauer or Chebyshev polynomials). Notable examples include the Rayleigh, Watson, and Bingham tests (García-Portugués et al., 2018, Fernández-de-Marcos et al., 2023).
• Projection-based Cramér–von Mises (CvM) tests compare the empirical c.d.f. of projections $\langle X_i, \gamma \rangle$ onto random directions $\gamma$ to the theoretical projection under uniformity, integrating discrepancies over all directions:

$$\text{CvM}_{n,q} = n \int_{S^q} \int_{-1}^1 [F_{n,\gamma}(x) - F_q(x)]^2\, dF_q(x)\, d\nu_q(\gamma)$$

with tractable U-statistic formulations and asymptotic distributions given by (possibly infinite) weighted sums of $\chi^2$ random variables (García-Portugués et al., 2020, Borodavka et al., 2023).

• Newer omnibus Sobolev tests employ kernels inspired by the von Mises–Fisher and Poisson family, featuring tuning parameters selected to maximize power against specific alternatives; null distributions are given via infinite series of weighted $\chi^2$s and efficient cross-validated selection schemes maintain calibration (Fernández-de-Marcos et al., 2023).

Recent work provides rigorous local efficiency properties under Bahadur efficiency theory and develops tractable numerical procedures for higher dimensions.

6. Dimensional Reduction and Transformation-Based Approaches

Transformation-based methods generalize the PIT to multivariate data, particularly via:

• Coordinatewise transformations for independent marginals: $u_{i,j} = F_j(x_{i,j})$.
• Normalizing flows for dependent or hierarchical models, fitting a flow to the data and transforming the sample to an approximate uniform distribution on $[0,1]^p$ (Shtembari et al., 2022).
• Volume-based reduction: mapping points to $v_i = \prod_{j=1}^p u_{i,j}$, with univariate uniformity of $v_i$ being a necessary condition for multivariate uniformity.

Uniformity across coordinates can be assessed via univariate tests (KS, CvM) on projections, with global $p$-values combined using min or product rules, exploiting the known joint distribution of min($p_1$, ..., $p_k$) for independent $p$-values (Shtembari et al., 2022).

7. Data Depth, Symmetry Reduction, and Ancillary Strategies

Tests based on data depth exploit the center-outward ordering provided by depth functions such as Tukey’s half-space or zonoid depth. Under uniformity, transformed depth values $F^{d}( D( X_i ) )$ are uniformly distributed on $[0,1]$, allowing application of standard univariate GoF tests. Distribution-free properties and consistency are ensured under mild assumptions on depth continuity and uniform convergence (Singh et al., 2021).

Symmetry-based reductions, as in (Klebanov et al., 2018), construct scalar random variables $S$ involving sample and reference sets (or two independent samples) such that uniformity holds if and only if the distribution of $S$ is symmetric about zero. Inequalities involving convex functions of the cumulative distribution functions of $S$ and its reflection form the basis for constructing consistent, high-dimensional two-sample tests.

Table: Summary of Main Classes of Multivariate Uniformity Tests

Methodology	Principle/Statistic	Domain/Context
Empirical Process & Brownian Sheets	$L^2$ norms of decomposed empirical processes	$[0,1]^p$ (hypercube)
Optimal Transport Ranks	Discrepancy in empirical char. functions of ranks	$\mathbb{R}^d$, general
Geometric/Spacing & Graph-based	Maximal spacing, edge-length functionals	Convex subsets $\subset \mathbb{R}^d$
Sobolev and Projection-based	Functionals of spherical harmonics; CvM projections	Hypersphere $S^q$
Transformation & PIT	Multivariate PIT to $[0,1]^p$, univariate reduction	Arbitrary, often $\mathbb{R}^d$
Depth-based	Uniformity of depth distributions; univariate GoF	$\mathbb{R}^d$, absolutely continuous

8. Implementation, Practical Considerations, and Performance

Practical implementation of modern multivariate uniformity tests often demands the solution of computationally challenging problems—e.g., optimal transport assignments, kernel density estimation, or evaluating high-dimensional U-statistics with complex kernels. In most cases, null distributions under the simple hypothesis can be simulated or are known explicitly (as for the Brownian sheet, Sobolev tests, or certain optimal transport tests).

Empirical studies and simulation evidence indicate:

• Tests based on Brownian sheet decompositions exhibit particularly high power against copula alternatives, with explicit control over the detection of dependence structures (Cabaña et al., 7 Sep 2025).
• Graph-based and spacing methods provide robust performance even in small samples and can detect local deviations, clusters, or contamination (Ebner et al., 2018).
• Sobolev and projection-based omnibus tests on the hypersphere are competitive against both unimodal and multimodal alternatives, and can be tuned for particular sensitivity profiles (Fernández-de-Marcos et al., 2023).
• Normalizing flow-based and transformation-based approaches offer flexible frameworks applicable to arbitrary parametric or generative models, facilitating goodness-of-fit and applied discovery/limit-setting problems (Shtembari et al., 2022).

9. Theoretical Innovations and Recent Advances

Recent research establishes uniform-over-dimension convergence theorems, such as uniform Lévy's continuity, providing essential conditions for family-wide consistency across both classical and high-dimensional regimes (Chowdhury et al., 24 Mar 2024). Optimal transport–based ranks and their distribution-free properties under the null, as well as the development of exact finite-sample null laws for certain test statistics, represent important methodological advances (Hlávka et al., 31 Jul 2025, Deb et al., 2021).

There is ongoing development in techniques for tuning parameters (oracle parameter selection, cross-validation) for kernel-based tests on the hypersphere, as well as in computational methods for handling high-dimensional integrals and random geometric graphs.

10. Applications and Future Directions

Applications of multivariate uniformity tests span simulation validation, spatial/point pattern analysis, astronomy (e.g., crater distribution studies), and statistical inference for Markov chain Monte Carlo output. The generality of transformation- and optimal transport–based approaches positions these methods for continued importance as high-dimensional, complex data structures become increasingly prevalent. Ongoing research emphasizes:

• The extension of distribution-free testing to broader settings (manifolds, graphs, networks).
• Efficient computation of test statistics and critical values in high dimensions.
• Rigorous performance assessment under various types of alternatives (e.g., dependence, scaling, multimodality).
• The integration of these methods into automated workflows for model checking and simulation-based inference.

The synthesis of analytic theory, computational advances, and practical diagnostics places the field of multivariate uniformity testing at the interface of probability, geometry, and modern data science.