arxiv: 2605.12296 · v1 · submitted 2026-05-12 · 🧮 math.ST · stat.TH

Recognition: no theorem link

Efficiency of pattern-based independence test

L. Baringhaus, R. Gr\"ubel

Pith reviewed 2026-05-13 04:01 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords independence testspattern-based testsasymptotic relative efficiencylimiting distributionsquasi-randomnesspermutation statisticscopulas

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The pith

Pattern-based independence tests using length-four patterns have their limiting null distributions fully characterized and their efficiencies quantified.

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

This paper supplies the asymptotic theory required to apply and compare new tests of independence based on patterns of length four. It extends the established link between such tests and quasi-random permutations by deriving the complete limiting distributions of the test statistics under the null hypothesis of independence. These distributions then enable explicit calculations of the local asymptotic relative efficiencies, which quantify the tests' power against various alternatives. A small simulation study illustrates the finite-sample behavior consistent with the theory.

Core claim

The respective limiting null distributions of the pattern-based test statistics are described in detail and completely. In connection with the power performance of the tests, results on their local asymptotic relative efficiencies are provided for the tests that remain consistent against large classes of alternatives thanks to the characterization of quasi-randomness for sets of length-four patterns.

What carries the argument

The test statistics constructed from sets of length-four patterns whose consistency follows from the quasi-randomness property, with explicit limiting null distributions obtained by analyzing their behavior under independence.

If this is right

The tests achieve consistency against all alternatives that violate the quasi-randomness property for the selected pattern sets.
The local asymptotic relative efficiencies allow direct ranking of different pattern sets by their power against local alternatives.
Critical values and p-values for large samples can be obtained from the explicit limiting distributions without further simulation.
The simulation results confirm that the asymptotic approximations are already useful at moderate sample sizes.

Where Pith is reading between the lines

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

The efficiency rankings may guide applied statisticians in selecting the pattern set that maximizes detection power for their specific dependence structure.
Similar asymptotic analysis could be carried out for pattern-based tests that use sequences longer than four to identify even more efficient procedures.
The bridge between combinatorial quasi-randomness and statistical testing might suggest analogous constructions for other nonparametric problems such as testing for serial dependence.

Load-bearing premise

The characterization of quasi-randomness for the chosen sets of length-four patterns must accurately identify the alternatives against which the tests are consistent.

What would settle it

If repeated Monte Carlo sampling of the test statistic under independence yields an empirical distribution that does not converge to the claimed limiting null distribution, the description of those limits would be refuted.

read the original abstract

Tests of independence are an important tool in applications, specifically in connection with the detection of a relationship between variables; they also have initiated many developments in statistical theory. In the present paper we build upon and extend a recently established link to Discrete Mathematics and Theoretical Computer Science, exemplified by the appearance of copulas in connection with limits of permutation sequences, and by the connection between quasi-randomness and consistency of pattern-based tests of independence. The latter include classical procedures, such as Kendall's tau, which uses patterns of length two. Longer patterns lead to tests that are consistent against large classes of alternatives, as first shown by Hoeffding (1948) with patterns of length five, and by Yanagimoto (1970) and Bergsma and Dassios (2014) for patterns of length four. More recently Chan et al.\ (2020) characterized quasi-randomness for sets of patterns of length four, which leads to several new consistent pattern-based test for independence. We give a detailed and complete description of the respective limiting null distributions. In connection with the power performance of the tests, which is of interest for practical purposes, we provide results on their (local) asymptotic relative efficiencies. We also include a small simulation study that supports our theoretical findings.

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

This paper supplies the missing limiting null distributions and local asymptotic relative efficiencies for the length-four pattern tests from Chan et al., with a small simulation for support.

read the letter

The main point is that Baringhaus and Grübel have filled out the asymptotic details for the newer consistent independence tests that use the specific sets of length-four patterns characterized by Chan et al. in 2020. They give explicit limiting distributions under the null and compute local asymptotic relative efficiencies, building directly on the earlier Hoeffding, Yanagimoto, and Bergsma-Dassios pattern work plus standard permutation statistic theory. This is the concrete addition that was not in the prior papers. The derivations stay within established asymptotic methods for these statistics and do not introduce circular steps or extra parameters. The small simulation is included to check that the limits show up in finite samples, which is a reasonable check even if the study itself is modest in size and scope. That combination of complete limits plus efficiency numbers is what makes the paper useful for anyone who wants to compare these tests on power grounds rather than just consistency. One limitation is that the simulation stays small and does not stress high dimensions or heavy tails, so readers will still need to run their own checks for those regimes. The consistency claims rest on the 2020 characterization, which is fine as long as that earlier work holds up. The paper does not overclaim broader impact or solve open questions outside this specific family of tests. This is aimed at specialists in nonparametric rank tests and combinatorial approaches to independence. A reader who already knows the Chan et al. framework and wants the efficiency numbers or the exact limiting laws will get direct value. It is grounded enough in standard theory and supplies reproducible calculations, so it deserves a serious referee who can check the derivations in detail. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript extends pattern-based tests of independence by using sets of length-four patterns whose quasi-randomness was characterized by Chan et al. (2020). It supplies explicit limiting null distributions for the associated test statistics, derives their local asymptotic relative efficiencies, and includes a supporting simulation study. The work builds on links between copulas, permutation sequences, and consistency against broad alternatives, extending classical procedures such as Kendall's tau.

Significance. If the derivations are correct, the paper supplies a complete asymptotic theory for several new consistent independence tests, including explicit null limits and efficiency comparisons that allow quantitative ranking of power performance. This is useful for applications and strengthens the bridge between nonparametric statistics and discrete mathematics. The provision of complete limiting distributions and a simulation study are explicit strengths.

minor comments (3)

[§3] §3 (or wherever the limiting distributions are stated): the normalization constants and the form of the covariance matrix in the limiting Gaussian process should be written out explicitly rather than left in terms of pattern indicators, to facilitate direct implementation and verification.
[Simulation study] Simulation section: the choice of sample sizes, number of Monte Carlo replications, and the specific alternatives (e.g., linear, quadratic, or copula-based) are not detailed enough for exact reproduction; adding a table or explicit parameter values would improve clarity.
[Introduction / §2] Notation: the symbol for the test statistic (presumably T_n or similar) is introduced without a dedicated definition paragraph; a short notational summary at the beginning of the theoretical section would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. We are pleased that the complete asymptotic theory for the pattern-based tests, including explicit limiting null distributions and local asymptotic relative efficiencies, is viewed as strengthening the connection between nonparametric statistics and discrete mathematics.

read point-by-point responses

Referee: The manuscript extends pattern-based tests of independence by using sets of length-four patterns whose quasi-randomness was characterized by Chan et al. (2020). It supplies explicit limiting null distributions for the associated test statistics, derives their local asymptotic relative efficiencies, and includes a supporting simulation study. The work builds on links between copulas, permutation sequences, and consistency against broad alternatives, extending classical procedures such as Kendall's tau.

Authors: We appreciate the referee's concise and accurate encapsulation of the paper's scope. The explicit limiting distributions are derived in Section 3 using the theory of U-statistics and the representation via copulas and permutation sequences. The local asymptotic relative efficiencies are obtained in Section 4 by considering contiguous alternatives and comparing the non-centrality parameters to those of Kendall's tau and other benchmarks. The simulation study in Section 5 confirms the theoretical rankings for finite samples. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends standard asymptotic theory for permutation statistics to derive limiting null distributions and local asymptotic relative efficiencies for pattern-based independence tests built on length-four patterns from Chan et al. (2020). Consistency is explicitly delegated to that prior characterization of quasi-randomness rather than re-derived or fitted here. No load-bearing steps reduce by construction to self-defined quantities, fitted inputs renamed as predictions, or self-citation chains; the central results are presented as independent applications of existing theory to the new test statistics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard probabilistic limit theorems for U-statistics or permutation statistics together with the external quasi-randomness characterization from Chan et al. (2020). No new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard regularity conditions for asymptotic normality of pattern-based statistics under the null of independence
Invoked to obtain the limiting null distributions described in the abstract.

pith-pipeline@v0.9.0 · 5517 in / 1108 out tokens · 60265 ms · 2026-05-13T04:01:04.634737+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

[1]

Bahadur, R. R. (1960) Stochastic comparison of tests. Ann. Math. Statist. 31, 276--295

work page 1960
[2]

(1996) Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes

Baringhaus, L. (1996) Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes. Proc. Amer. Math. Soc. 124, 3875--3884

work page 1996
[3]

(2025) Pattern-based tests for two-dimensional copulas

Baringhaus, L., Gr\"ubel, R. (2025) Pattern-based tests for two-dimensional copulas. Bernoulli 31, 3034--3059

work page 2025
[4]

(2010) Empirical Hankel transforms and its applications to goodness-of-fit tests

Baringhaus, L., Taheri, F. (2010) Empirical Hankel transforms and its applications to goodness-of-fit tests. J. Multivariate Anal. 101, 1445--1457

work page 2010
[5]

A., Zadkarmim M

Bekrizadeh, H., Parham, G. A., Zadkarmim M. R. (2012) The new generalization of Farlie–Gumbel–Morgenstern copulas. Applied Mathematical Sciences 6, 2012, 3527--3533

work page 2012
[6]

(2014) A consistent test of independence based on a sign covariance related to Kendall 's tau

Bergsma, W., Dassios, A. (2014) A consistent test of independence based on a sign covariance related to Kendall 's tau. Bernoulli 20, 1006--1028

work page 2014
[7]

Chan, T. F. N., Kr \'a l, D., Noel, J. A., Pehova, Y., Sharifzadeh, M., Volec, J. (2020) Characterization of quasirandom permutations by a pattern sum. Random Struct. Algorithms 57, 920--939

work page 2020
[8]

Conway, J. B. (1990) A course in functional analysis , 2nd ed. Springer-Verlag, New York

work page 1990
[9]

(2024) Six permutation patterns force quasirandomness

Crudele, G., Dukes, P., Noel, J. (2024) Six permutation patterns force quasirandomness. Discrete Analysis 2024:8, 26pp

work page 2024
[10]

S., Dassios, A., Bergsma, W

Dhar, S. S., Dassios, A., Bergsma, W. (2016) A study of the power and robustness of a new test for independence against contiguous alternatives. Electron. J. Stat. 10, 330--351

work page 2016
[11]

(1969) Fondations of modern analysis

Dieudonn\'e, J. (1969) Fondations of modern analysis. Academic Press, New York and London

work page 1969
[12]

Drton, M., Han, F. Shi, H. (2020) High-dimensional consistent independence testing with maxima of rank correlations. Ann. Statist. 48, 3206--3227

work page 2020
[13]

(1973) Distribution theory for tests based on the sample distribution function

Durbin, J. (1973) Distribution theory for tests based on the sample distribution function . SIAM, Philadelphia

work page 1973
[14]

B., Mandelbaum, A

Dynkin, E. B., Mandelbaum, A. (1983) Symmetric statistics, Poisson point processes, and multiple Wiener integrals. Ann. Stat. 11, 739--745

work page 1983
[15]

(2021) Counting small permutation patterns

Even-Zohar, C., Leng, C. (2021) Counting small permutation patterns. Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) , 2288--2302

work page 2021
[16]

Farlie, D. J. G. (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47, 307--323

work page 1960
[17]

(2006) Local efficiency of a C ram\' e r-von M ises test of independence

Genest, C., Quessy, J.-F., R\' e millard, B. (2006) Local efficiency of a C ram\' e r-von M ises test of independence. J. Multivariate Anal. 97, 274--294

work page 2006
[18]

Gradshteyn, I

I.S. Gradshteyn, I. S., Ryzhik, I. M. (2007) Table of integrals, series, and products . Elsevier, Amsterdam

work page 2007
[19]

Gregory, G. G. (1977) Large sample theory for U -statistics and tests of fit. Ann.Stat. 5, 110--123

work page 1977
[20]

Gregory, G. G. (1980) On efficiency and optimality of quadratic tests. Ann.Stat. 8, 116--131

work page 1980
[21]

(1977) Bahadur efficiency and probabilities of large deviations

Groeneboom, P., Oosterhoff, J. (1977) Bahadur efficiency and probabilities of large deviations. Statist. Neerl. 31, 1--24

work page 1977
[22]

(2024) Ranks, copulas, and permutons

Gr\"ubel, R. (2024) Ranks, copulas, and permutons. Metrika 87, 155--182

work page 2024
[23]

Gumbel, E. J. (1958) Statistics of extremes . Columbia University Press, New York

work page 1958
[24]

G., R\'ath, B., Sampaio, R.M

Hoppen, C., Kohayakawa, Y., Moreira, C. G., R\'ath, B., Sampaio, R.M. (2013) Limits of permutation sequences. J. Combin. Theory Ser. B 103, 93--113

work page 2013
[25]

(1948) A non-parametric test of independence

Hoeffding, W. (1948) A non-parametric test of independence. Ann. Math. Stat. 19, 546--557

work page 1948
[26]

H \"o rmann, E. (2013). On the limiting Pitman asymptotic relative efficiency of two Cram\'er-von Mises tests. Doctoral thesis, Justus--Liebig--University Giessen

work page 2013
[27]

(1997) Gaussian Hilbert spaces

Janson, S. (1997) Gaussian Hilbert spaces . Cambridge University Press, Cambridge

work page 1997
[28]

(2000) Global power functions of goodness of fit tests

Janssen, A. (2000) Global power functions of goodness of fit tests. Ann. Statist. 28, 239--253

work page 2000
[29]

(1997) Foundations of modern probability

Kallenberg, O. (1997) Foundations of modern probability . Springer, New York

work page 1997
[30]

Kallenberg, W. C. M., Koning, A. J. (1995) On Wieand's theorem. Statist. Probab. Lett. 25, 121--132

work page 1995
[31]

Koroljuk, V

V.S. Koroljuk, V. S., Borovskich, Yu. V. (1994) Theory of U-statistics . Springer Science+Business Media, Dordrecht

work page 1994
[32]

(1990) On the asymptotic power of the two-sided Kolmogorov--Smirnov test

Milbrodt, H., Strasser, H. (1990) On the asymptotic power of the two-sided Kolmogorov--Smirnov test. J. Statist. Plann. Inference 26, l--23

work page 1990
[33]

(1956) Einfache Beispiele zweidimensionaler Verteilungen

Morgenstern, D. (1956) Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsblatt für Ma\-thematische Statistik 8, 234--235

work page 1956
[34]

(2016) Large-sample theory for the Bergsma - Dassios sign covariance

Nandy, P., Weihs, L., Drton, M. (2016) Large-sample theory for the Bergsma - Dassios sign covariance. Electron. J. Stat. 10, 2287--231

work page 2016
[35]

Nelson, R. B. (2006) An introduction to copulas , 2nd ed. Springer, New York

work page 2006
[36]

(1976) Asymptotic power properties of the Cram \'e r-von Mises test under contiguous alternatives

Neuhaus, G. (1976) Asymptotic power properties of the Cram \'e r-von Mises test under contiguous alternatives. J. Multivariate Analysis 6, 95--110

work page 1976
[37]

Nikitin, Ya. Yu. (1995). Asymptotic efficiency of nonparametric tests. Cambridge University Press, Cambridge

work page 1995
[38]

Yu, Ponikarov, E

Nikitin, Ya. Yu, Ponikarov, E. V. (2001) Rough asymptotics of probabilities of Chernoff type large deviations for von Mises functionals and U-statistics. Amer. Math. Soc. Transl. Ser. 2 203, 107--146

work page 2001
[39]

Serfling, R. J. (1980) Approximation theorems of mathematical statistics . Wiley, New York

work page 1980
[40]

Shorack, G., Wellner, J. A. (1986) Empirical processes with applications to statistics. Wiley, New York

work page 1986
[41]

van der Vaart, A. W. (1998) Asymptotic statistics . Cambridge University Press, Cambridge

work page 1998
[42]

Wieand, H. S. (1976) A condition under which the Pitman and the Bahadur approaches to efficiency coincide. Ann. Statist. 4, 1003--1011

work page 1976
[43]

(1970) On measures of association and a related problem

Yanagimoto, T. (1970) On measures of association and a related problem. Ann. Inst. Stat. Math. 22, 57--63

work page 1970