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

Alejandra Caba\~na; Enrique M. Caba\~na

arxiv: 2605.17474 · v1 · pith:UXUES5S4new · submitted 2026-05-17 · 🧮 math.ST · stat.ME· stat.TH

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

Alejandra Caba\~na , Enrique M. Caba\~na This is my paper

Pith reviewed 2026-05-19 22:32 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH

keywords multivariate goodness-of-fitempirical distribution functionBrownian sheetuniformity testmultivariate normalityspherical symmetryindependence testprobability integral transform

0 comments

The pith

New EDF tests reduce multivariate normality, symmetry, and independence checks to uniformity testing on the unit hypercube via Brownian sheet decomposition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends two existing goodness-of-fit procedures, the m-test and s-test, originally defined for the unit hypercube, to a wider set of multivariate problems. It shows that many standard null hypotheses in multivariate statistics can be mapped to a uniformity hypothesis on [0,1]^p by applying componentwise probability integral transforms. The resulting tests rely on a decomposition of the Brownian sheet into independent processes together with a decomposition of signed measures that isolates coordinate interactions. These constructions are then used to produce explicit procedures for testing uniformity on the sphere, multivariate normality, spherical and elliptical symmetry, and mutual independence. A reader would care because the same underlying machinery supplies a single family of tests that handles several common multivariate checks while remaining sensitive to dependence patterns that affect coordinates jointly.

Core claim

Whenever a null hypothesis implies that the distribution factorizes into independent continuous components after a suitable mapping, the testing problem reduces to a uniformity test on the hypercube; the m-test and s-test, built from the independent Gaussian processes obtained by deconstructing the p-parameter Brownian sheet, can therefore be applied directly after the transform. The paper further exploits the decomposition of finite signed measures into zero-marginal components to isolate interactions among coordinates, yielding new explicit procedures for uniformity on the hypersphere, multivariate normality, spherical and elliptical symmetry, and independence. Power comparisons indicate a

What carries the argument

The m-test and s-test constructed from the deconstruction of the p-parameter Brownian sheet into independent Gaussian processes, together with the decomposition of finite signed measures into zero-marginal components.

If this is right

The same reduction applies to testing uniformity on the hypersphere after a radial transformation.
Multivariate normality can be tested by first transforming to uniformity on the hypercube and then applying the m-test or s-test.
Spherical and elliptical symmetry hypotheses are handled by the same measure-decomposition step that isolates coordinate interactions.
Independence among coordinates is detected by the tests' sensitivity to non-zero cross terms after the marginal transforms.
The procedures remain valid for any dimension p because the Brownian-sheet deconstruction scales with the number of coordinates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mapping-and-decomposition strategy could be tried on other null hypotheses whose joint law admits a factorization after a monotone transform, such as certain copula families.
Because the tests isolate coordinate interactions explicitly, they may offer an advantage in high-dimensional settings where dependence is sparse across pairs.
Extending the finite signed-measure decomposition to censored or truncated data would constitute a natural next step not addressed in the paper.
The observed sensitivity to joint dependence structures suggests the tests could be useful as diagnostic tools inside multivariate modeling pipelines.

Load-bearing premise

A null hypothesis must allow the joint distribution to factorize into independent continuous marginals after a suitable transformation so that the problem reduces to uniformity testing on the hypercube.

What would settle it

A simulation study in which the new tests exhibit systematically lower power than established competitors (for example, against coordinate-wise dependence alternatives that preserve marginal normality) would contradict the claim of competitive sensitivity.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the m-test/s-test Brownian sheet framework to uniformity on the sphere, multivariate normality, symmetry, and independence via PIT reductions, with decent empirical power results, but the key step of preserving the independent Gaussian process decomposition under parameter estimation and non-cube supports is asserted rather than derived in detail.

read the letter

The core contribution is applying the existing deconstruction of the p-dimensional Brownian sheet into independent processes to four new testing problems: uniformity on the hypersphere, multivariate normality, spherical and elliptical symmetry, and independence. They reduce each to a uniformity test on the hypercube by componentwise probability integral transforms after a suitable mapping, then reuse the m-test and s-test statistics based on the signed-measure decomposition that isolates coordinate interactions. This is genuinely new for these specific extensions, and the empirical power comparisons are a concrete plus—they indicate the procedures pick up coordinate-wise and joint dependencies at least as well as standard competitors in the simulations shown. The unified treatment across these hypotheses is a practical strength if the technical reduction holds. The soft spot is exactly the one the stress-test flags. For the simple uniformity case on the cube the independence of the component processes is built in, but normality and elliptical symmetry involve estimated parameters, and the sphere has curved support. The paper states that the mapping leaves the null distribution unchanged, yet I did not find an explicit check that the finite signed-measure decomposition and the resulting independent-GP structure survive either the estimation step or the non-rectangular domain. If that independence breaks, the critical values derived from the original deconstruction no longer apply directly. The power tables are useful but would be stronger with more detail on how the competing procedures were calibrated and whether the sample sizes match typical applied settings. This work is aimed at researchers in nonparametric multivariate testing who already know EDF methods or Brownian-sheet constructions; it is not a broad reorganization of practice but a targeted addition to the toolkit. A serious referee should be asked to verify the preservation argument for the composite cases and to confirm the power-study implementation. I would send it out rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper extends the m-test and s-test for uniformity on the hypercube [0,1]^p, which are based on a deconstruction of the p-parameter Brownian sheet into independent Gaussian processes, to testing uniformity on the hypersphere S^p, multivariate normality, spherical and elliptical symmetry, and independence in R^p. The extensions rely on reducing these problems to hypercube uniformity tests via componentwise probability integral transforms after suitable (possibly parameter-dependent) mappings, with the finite signed-measure decomposition used to isolate coordinate interactions. Empirical power comparisons are presented to show that the resulting procedures are competitive with existing methods, with particular sensitivity to coordinate-based and joint dependencies.

Significance. If the reductions preserve the independent-GP structure and null distribution of the test statistics under composite hypotheses, the work would provide a unified, EDF-based framework with a novel deconstruction that could be useful for detecting specific dependence structures. The empirical results, if reproducible and correctly calibrated, would support practical competitiveness, but the significance hinges on rigorous justification of the mapping step rather than assertion alone.

major comments (2)

[Introduction and extensions sections (around the statement of the reduction principle)] The central reduction for composite hypotheses (multivariate normality, elliptical symmetry, independence) is asserted in the abstract and introduction via 'suitable mapping' and componentwise PIT, but no explicit verification appears that the independence of the component Gaussian processes from the Brownian-sheet deconstruction survives parameter estimation; if the finite signed-measure decomposition is altered, the null distribution of the m-test/s-test statistics is no longer the one derived for the hypercube case.
[Section on hypersphere uniformity and spherical symmetry] For uniformity on the hypersphere S^p and spherical symmetry, the non-rectangular support means the transformed variables are not supported on the full hypercube; the paper does not show that the zero-marginal decomposition and resulting test statistic remain distributionally equivalent to the original hypercube construction under the null.

minor comments (2)

[Methodology] Clarify the exact definitions of the extended test statistics (e.g., how the signed measures are constructed post-mapping) with explicit formulas or pseudocode.
[Empirical study] Power tables should include the exact sample sizes, number of Monte Carlo replications, and parameter estimation methods used for the composite cases to allow direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We address each major comment below and have made revisions to clarify the theoretical justifications for the reductions under composite hypotheses and for the hypersphere case. These changes enhance the rigor of the presentation without altering the core contributions.

read point-by-point responses

Referee: [Introduction and extensions sections (around the statement of the reduction principle)] The central reduction for composite hypotheses (multivariate normality, elliptical symmetry, independence) is asserted in the abstract and introduction via 'suitable mapping' and componentwise PIT, but no explicit verification appears that the independence of the component Gaussian processes from the Brownian-sheet deconstruction survives parameter estimation; if the finite signed-measure decomposition is altered, the null distribution of the m-test/s-test statistics is no longer the one derived for the hypercube case.

Authors: We agree that explicit verification strengthens the argument for composite hypotheses. The manuscript relies on the principle that a suitable mapping followed by componentwise PIT yields independent uniforms under the null, to which the hypercube deconstruction applies directly. To address parameter estimation, the revised manuscript adds an asymptotic argument in a new subsection: consistent estimators and continuous PIT ensure the transformed processes converge in distribution to the same independent Gaussian processes derived for the simple case, so the finite signed-measure zero-marginal decomposition is preserved in the limit. For finite-sample critical values we recommend parametric bootstrap, and we have added a small simulation confirming empirical sizes track nominal levels. This approach maintains the practical utility of the m-test and s-test while acknowledging the composite setting. revision: partial
Referee: [Section on hypersphere uniformity and spherical symmetry] For uniformity on the hypersphere S^p and spherical symmetry, the non-rectangular support means the transformed variables are not supported on the full hypercube; the paper does not show that the zero-marginal decomposition and resulting test statistic remain distributionally equivalent to the original hypercube construction under the null.

Authors: We thank the referee for this observation on support. For uniformity on S^p we use the inverse of the marginal cdfs induced by the uniform measure on the sphere (or equivalent spherical-coordinate mappings) followed by componentwise PIT. Although the original variables lie on a lower-dimensional manifold, the resulting PIT-transformed variables are exactly independent Uniform[0,1] under the null. The revised manuscript includes a short appendix proof showing that the marginals remain uniform, so the finite signed-measure decomposition into zero-marginal components is unchanged and the null distribution of the test statistics coincides with the hypercube case. We have also clarified the explicit mapping in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reductions rest on standard probability facts

full rationale

The paper extends its prior m-test/s-test construction for the hypercube by invoking the standard fact that a null hypothesis factorizing into independent continuous components after a suitable mapping reduces to uniformity testing via componentwise PITs. This is asserted as a known property of continuous distributions rather than derived from the paper's own fitted quantities or self-referential equations. No load-bearing step equates the final test statistic or its null distribution to a parameter fit on the evaluation data, nor does any self-citation chain substitute for an independent verification of the independence-preserving property under composite hypotheses. Empirical power comparisons supply external content. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of a unique deconstruction of the p-parameter Brownian sheet into independent Gaussian processes (taken from prior work) and on the standard fact that continuous marginal cdfs yield uniform transforms under the null.

axioms (1)

domain assumption A null hypothesis that factorizes into independent continuous components after a suitable mapping reduces to a uniformity test on the hypercube via componentwise PITs.
Invoked in the abstract to justify reduction of normality, symmetry, and independence problems to the hypercube case.

pith-pipeline@v0.9.0 · 5716 in / 1253 out tokens · 37461 ms · 2026-05-19T22:32:18.682872+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the standard p-parametric Gaussian Wiener measure W on [0,1]^p can be decomposed into a linear combination of independent Brownian pillows b_H

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Caba˜ na, E

A. Caba˜ na, E. M. Caba˜ na, Brownian sheet and uniformity tests on the hypercube, to appear in Statistica.arXiv:2509.06134. URLhttps://arxiv.org/abs/2509.06134

work page arXiv
[2]

Caba˜ na, E

A. Caba˜ na, E. M. Caba˜ na, Multivariate tests of uniformity, normality, sym- metry and independence, r package, version 0.2.0. URLhttps://github.com/emcabana/MuniCandS

work page
[3]

Durbin, Distribution Theory for Tests Based on the Sample Distribu- tion Function, no

J. Durbin, Distribution Theory for Tests Based on the Sample Distribu- tion Function, no. 9 in CBMS-NSF Regional Conference Series in Ap- plied Mathematics, Society for Industrial and Applied Mathematics, 1973. doi:10.1137/1.9781611970568

work page doi:10.1137/1.9781611970568 1973
[4]

Banerjee, I

A. Banerjee, I. S. Dhillon, J. Ghosh, S. Sra, Clustering on the unit hyper- sphere using von mises–fisher distributions, Journal of Machine Learning Research 6 (Sep) (2005) 1345–1382

work page 2005
[5]

URLhttps://www.R-project.org/

R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2025). URLhttps://www.R-project.org/

work page 2025
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Garc´ ıa-Portugu´ es, T

E. Garc´ ıa-Portugu´ es, T. Verdebout, sphunif: Uniformity Tests on the Cir- cle, Sphere, and Hypersphere (2025). URLhttps://cran.r-project.org/package=sphunif

work page 2025
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Garc´ ıa-Portugu´ es, T

E. Garc´ ıa-Portugu´ es, T. Verdebout, An overview of uniformity tests on the hypersphere, preprint (2018)

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Baringhaus, N

L. Baringhaus, N. Henze, A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1) (1988) 339–348. doi:10.1007/BF02613322

work page doi:10.1007/bf02613322 1988
[9]

Henze, B

N. Henze, B. Zirkler, A class of invariant consistent tests for multivariate normality, Communications in Statistics - Theory and Methods 19 (10) (1990) 3595–3617.doi:10.1080/03610929008830400. 23

work page doi:10.1080/03610929008830400 1990
[10]

D¨ orr, B

P. D¨ orr, B. Ebner, N. Henze, A new test of multivariate normality by a double estimation in a characterizing pde, Metrika 84 (3) (2021) 401–427. doi:10.1007/s00184-020-00795-x

work page doi:10.1007/s00184-020-00795-x 2021
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G. J. Sz´ ekely, M. L. Rizzo, A new test for multivariate normality, Journal of Multivariate Analysis 93 (1) (2005) 58–80.doi:10.1016/j.jmva.2003. 12.002

work page doi:10.1016/j.jmva.2003 2005
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J. A. Doornik, H. Hansen, An omnibus test for univariate and multivariate normality, Oxford Bulletin of Economics and Statistics 70 (s1) (2008) 927– 939.doi:10.1111/j.1468-0084.2008.00537.x

work page doi:10.1111/j.1468-0084.2008.00537.x 2008
[13]

Royston, Approximating the shapiro-wilk w-test for non-normality, Jour- nal of the Royal Statistical Society: Series D (The Statistician) 41 (1) (1992) 113–121

P. Royston, Approximating the shapiro-wilk w-test for non-normality, Jour- nal of the Royal Statistical Society: Series D (The Statistician) 41 (1) (1992) 113–121

work page 1992
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J. A. Villase˜ nor-Alva, E. Gonz´ alez-Estrada, A generalization of shapiro- wilk’s test for multivariate normality, Communications in Statistics— Theory and Methods 38 (11) (2009) 1870–1883.doi:10.1080/ 03610920802474465

work page 2009
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Butsch, B

L. Butsch, B. Ebner, mnt: Affine Invariant Tests of Multivariate Normality, r package version 1.3 (2020).doi:10.32614/CRAN.package.mnt

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Banerjee, A

B. Banerjee, A. K. Ghosh, A nonparametric test of spherical symmetry applicable to high dimensional data (2024).arXiv:2403.12491,doi:10. 48550/arXiv.2403.12491

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Manzotti, F

A. Manzotti, F. J. P´ erez, A. J. Quiroz, A statistic for testing the null hypothesis of elliptical symmetry, Journal of Multivariate Analysis 81 (2) (2002) 274–285.doi:10.1006/jmva.2001.2007

work page doi:10.1006/jmva.2001.2007 2002
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J. R. Schott, Testing for elliptical symmetry in covariance-matrix-based analyses, Statistics & Probability Letters 60 (4) (2002) 395–404.doi: 10.1016/S0167-7152(02)00306-1

work page doi:10.1016/s0167-7152(02)00306-1 2002
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Azzalini, A class of distributions which includes the normal ones, Scan- dinavian Journal of Statistics 12 (2) (1985) 171–178

A. Azzalini, A class of distributions which includes the normal ones, Scan- dinavian Journal of Statistics 12 (2) (1985) 171–178

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Hallin, S

M. Hallin, S. G. Meintanis, K. Nordhausen, Consistent distribution free affine invariant tests for the validity of independent component models, arXiv preprint arXiv:2404.07632 (2024).arXiv:2404.07632. URLhttps://arxiv.org/abs/2404.07632

work page arXiv 2024
[21]

B. B¨ ottcher, Dependence and dependence structures: Estimation and visu- alization using the unifying concept of distance multivariance, Open Statis- tics 1 (1) (2020) 1–30.doi:10.1515/stat-2020-0001

work page doi:10.1515/stat-2020-0001 2020
[22]

B¨ ucher, C

A. B¨ ucher, C. Pakzad, Testing for independence in high dimensions based on empirical copulas, The Annals of Statistics 52 (1) (2024) 311–334.doi: 10.1214/23-AOS2348. 24

work page doi:10.1214/23-aos2348 2024

[1] [1]

Caba˜ na, E

A. Caba˜ na, E. M. Caba˜ na, Brownian sheet and uniformity tests on the hypercube, to appear in Statistica.arXiv:2509.06134. URLhttps://arxiv.org/abs/2509.06134

work page arXiv

[2] [2]

Caba˜ na, E

A. Caba˜ na, E. M. Caba˜ na, Multivariate tests of uniformity, normality, sym- metry and independence, r package, version 0.2.0. URLhttps://github.com/emcabana/MuniCandS

work page

[3] [3]

Durbin, Distribution Theory for Tests Based on the Sample Distribu- tion Function, no

J. Durbin, Distribution Theory for Tests Based on the Sample Distribu- tion Function, no. 9 in CBMS-NSF Regional Conference Series in Ap- plied Mathematics, Society for Industrial and Applied Mathematics, 1973. doi:10.1137/1.9781611970568

work page doi:10.1137/1.9781611970568 1973

[4] [4]

Banerjee, I

A. Banerjee, I. S. Dhillon, J. Ghosh, S. Sra, Clustering on the unit hyper- sphere using von mises–fisher distributions, Journal of Machine Learning Research 6 (Sep) (2005) 1345–1382

work page 2005

[5] [5]

URLhttps://www.R-project.org/

R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2025). URLhttps://www.R-project.org/

work page 2025

[6] [6]

Garc´ ıa-Portugu´ es, T

E. Garc´ ıa-Portugu´ es, T. Verdebout, sphunif: Uniformity Tests on the Cir- cle, Sphere, and Hypersphere (2025). URLhttps://cran.r-project.org/package=sphunif

work page 2025

[7] [7]

Garc´ ıa-Portugu´ es, T

E. Garc´ ıa-Portugu´ es, T. Verdebout, An overview of uniformity tests on the hypersphere, preprint (2018)

work page 2018

[8] [8]

Baringhaus, N

L. Baringhaus, N. Henze, A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1) (1988) 339–348. doi:10.1007/BF02613322

work page doi:10.1007/bf02613322 1988

[9] [9]

Henze, B

N. Henze, B. Zirkler, A class of invariant consistent tests for multivariate normality, Communications in Statistics - Theory and Methods 19 (10) (1990) 3595–3617.doi:10.1080/03610929008830400. 23

work page doi:10.1080/03610929008830400 1990

[10] [10]

D¨ orr, B

P. D¨ orr, B. Ebner, N. Henze, A new test of multivariate normality by a double estimation in a characterizing pde, Metrika 84 (3) (2021) 401–427. doi:10.1007/s00184-020-00795-x

work page doi:10.1007/s00184-020-00795-x 2021

[11] [11]

G. J. Sz´ ekely, M. L. Rizzo, A new test for multivariate normality, Journal of Multivariate Analysis 93 (1) (2005) 58–80.doi:10.1016/j.jmva.2003. 12.002

work page doi:10.1016/j.jmva.2003 2005

[12] [12]

J. A. Doornik, H. Hansen, An omnibus test for univariate and multivariate normality, Oxford Bulletin of Economics and Statistics 70 (s1) (2008) 927– 939.doi:10.1111/j.1468-0084.2008.00537.x

work page doi:10.1111/j.1468-0084.2008.00537.x 2008

[13] [13]

Royston, Approximating the shapiro-wilk w-test for non-normality, Jour- nal of the Royal Statistical Society: Series D (The Statistician) 41 (1) (1992) 113–121

P. Royston, Approximating the shapiro-wilk w-test for non-normality, Jour- nal of the Royal Statistical Society: Series D (The Statistician) 41 (1) (1992) 113–121

work page 1992

[14] [14]

J. A. Villase˜ nor-Alva, E. Gonz´ alez-Estrada, A generalization of shapiro- wilk’s test for multivariate normality, Communications in Statistics— Theory and Methods 38 (11) (2009) 1870–1883.doi:10.1080/ 03610920802474465

work page 2009

[15] [15]

Butsch, B

L. Butsch, B. Ebner, mnt: Affine Invariant Tests of Multivariate Normality, r package version 1.3 (2020).doi:10.32614/CRAN.package.mnt

work page doi:10.32614/cran.package.mnt 2020

[16] [16]

Banerjee, A

B. Banerjee, A. K. Ghosh, A nonparametric test of spherical symmetry applicable to high dimensional data (2024).arXiv:2403.12491,doi:10. 48550/arXiv.2403.12491

work page arXiv 2024

[17] [17]

Manzotti, F

A. Manzotti, F. J. P´ erez, A. J. Quiroz, A statistic for testing the null hypothesis of elliptical symmetry, Journal of Multivariate Analysis 81 (2) (2002) 274–285.doi:10.1006/jmva.2001.2007

work page doi:10.1006/jmva.2001.2007 2002

[18] [18]

J. R. Schott, Testing for elliptical symmetry in covariance-matrix-based analyses, Statistics & Probability Letters 60 (4) (2002) 395–404.doi: 10.1016/S0167-7152(02)00306-1

work page doi:10.1016/s0167-7152(02)00306-1 2002

[19] [19]

Azzalini, A class of distributions which includes the normal ones, Scan- dinavian Journal of Statistics 12 (2) (1985) 171–178

A. Azzalini, A class of distributions which includes the normal ones, Scan- dinavian Journal of Statistics 12 (2) (1985) 171–178

work page 1985

[20] [20]

Hallin, S

M. Hallin, S. G. Meintanis, K. Nordhausen, Consistent distribution free affine invariant tests for the validity of independent component models, arXiv preprint arXiv:2404.07632 (2024).arXiv:2404.07632. URLhttps://arxiv.org/abs/2404.07632

work page arXiv 2024

[21] [21]

B. B¨ ottcher, Dependence and dependence structures: Estimation and visu- alization using the unifying concept of distance multivariance, Open Statis- tics 1 (1) (2020) 1–30.doi:10.1515/stat-2020-0001

work page doi:10.1515/stat-2020-0001 2020

[22] [22]

B¨ ucher, C

A. B¨ ucher, C. Pakzad, Testing for independence in high dimensions based on empirical copulas, The Annals of Statistics 52 (1) (2024) 311–334.doi: 10.1214/23-AOS2348. 24

work page doi:10.1214/23-aos2348 2024