The post-hoc test for local dependence

Bart{\l}omiej Gibas; Bogdan \'Cmiel

arxiv: 2512.20280 · v2 · pith:3VQPTFQPnew · submitted 2025-12-23 · 📊 stat.ME

The post-hoc test for local dependence

Bogdan \'Cmiel , Bart{\l}omiej Gibas This is my paper

Pith reviewed 2026-05-16 20:37 UTC · model grok-4.3

classification 📊 stat.ME

keywords independence testinglocal dependencequantile dependence functioncritical surfacescopulapost-hoc analysisstatistical significance

0 comments

The pith

A test based on the quantile dependence function uses critical surfaces to detect local dependence while preserving the global significance level.

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

The paper develops a method that tests independence both globally and locally by working with the quantile dependence function from copula theory. Instead of comparing a single statistic to one threshold, it constructs critical surfaces so that the chance of exceeding the surface at any local point equals a fixed small probability when the variables are truly independent. This setup lets users see exactly where dependence appears, measure its local strength, and display the results graphically without raising the overall false-positive rate. A reader would care because most existing tests only answer whether any dependence exists at all, leaving the location and pattern of dependence unknown.

Core claim

Relying on copula-based results, the authors introduce a novel method for testing global and local statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold, they introduce critical surfaces that guarantee a locally equal probability of exceeding them under independence. This enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.

What carries the argument

Critical surfaces defined on the quantile dependence function that deliver equal local exceedance probability under independence.

If this is right

Global independence is rejected only if at least one local region exceeds its surface, keeping the family-wise error controlled.
Local regions of dependence can be identified and ranked by the amount they exceed their surface.
Graphical plots of the surfaces and observed values make the locations and strength of dependence directly visible.
The same framework supplies both a global p-value and a map of significant local departures.

Where Pith is reading between the lines

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

The surfaces could be adapted to time-series or spatial data by replacing the copula with an appropriate dependence measure.
In high-dimensional settings the local map might help select which variable pairs drive overall dependence.
The method may reduce the need for separate post-hoc procedures after a global test rejects.

Load-bearing premise

The copula-derived quantile dependence function correctly captures the dependence structure, and the critical surfaces can be built to give exactly the same local exceedance rate everywhere when variables are independent.

What would settle it

Simulate independent data, apply the test, and check whether the proportion of local points exceeding their critical surface stays exactly at the nominal level across repeated trials.

Figures

Figures reproduced from arXiv: 2512.20280 by Bart{\l}omiej Gibas, Bogdan \'Cmiel.

**Figure 1.** Figure 1: Histograms of 106 Monte Carlo realizations for q¯n and qˆn (k = ⌊ √ n⌋) based on 100 samples from uniform distribution on the interval (0, 1) in points (from upper-left): (0.05, 0.05), (0.45, 0.45). where qn(0, v) = qn(1, v) = qn(u, 0) = qn(u, 1) = 0. It is worth to notice that within each sub-rectangle of grid, the estimator Cn remains constant. Therefore qn is fully determined by the points (u, v) inside… view at source ↗

**Figure 2.** Figure 2: Critical surfaces (from the left): upper [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: The scatter plot presents studentized residuals (y-axis) against fitted values from linear regres [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: The scatter plot shows the data after cdf transformation: contents (y-axis) and buildings [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

read the original abstract

The concept of independence plays a crucial role in probability theory and has been the subject of extensive research in recent years. Numerous approaches have been proposed to test for independence; however, most of them address the problem only at a global level. From a practical perspective, it is important not only to determine whether the data are dependent but also to identify where this dependence occurs and how strong it is. The graphical presentation of results is another essential aspect that should not be neglected, as it considerably enhances interpretability. The main objective of this work is to propose a solution that considers these aspects simultaneously. Relying on copula-based results, we introduce a novel method for testing global and local statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold and subsequently deciding whether to reject the independence hypothesis, we introduce so-called critical surfaces that guaranty a locally equal probability of exceeding them under independence. This approach enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.

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 critical surfaces idea for local dependence testing is a reasonable extension of copula methods, but exact finite-sample local uniformity looks like the part that needs the most checking.

read the letter

The paper's main move is to replace a single global threshold with critical surfaces on the quantile dependence function. Under independence these surfaces are supposed to be crossed with exactly the same probability at every local point while the overall rejection rate stays at alpha. That framing lets you both test global independence and point to specific regions where dependence shows up, which is the practical gap they are trying to fill. The graphical angle is also a plus for interpretability in applied settings. They build on existing copula results for the quantile dependence process, so the foundation is not invented from scratch. If the surfaces can be constructed without extra tuning parameters that depend on unknown marginals, that would be a clean contribution. The soft spot is the exactness claim. The stress-test note is right to flag that limiting-distribution arguments usually give only asymptotic local calibration; for finite n the exceedance probabilities across the grid will typically drift unless the surfaces are either closed-form and parameter-free or calibrated by simulation that itself does not leak information about the data. The abstract says the surfaces “guaranty” local equality, but without seeing the actual construction or the simulation checks it is hard to judge how close they get in practice. This is the kind of work that belongs in a reading group for people who already use copula-based dependence tests and want local diagnostics. It is not yet ready for routine use until the finite-sample behavior is shown clearly. I would send it to referees rather than desk-reject; the idea is coherent enough and the gap it targets is real, even if the calibration details probably need revision.

Referee Report

1 major / 1 minor

Summary. The paper proposes a post-hoc test for local dependence based on the quantile dependence function from copula theory. Instead of a single global threshold, it constructs critical surfaces that are claimed to deliver exactly equal local exceedance probabilities at every point under the null of independence while preserving the global type-I error rate at level alpha. This is intended to enable both global testing and interpretable local identification of dependence regions with graphical output.

Significance. If the critical surfaces achieve the stated exact local calibration in finite samples, the method would offer a useful advance for dependence diagnostics that require both global control and local resolution. The copula-based framing is standard in the field, but the practical value hinges on whether the local uniformity holds beyond asymptotics and whether the surfaces can be computed without additional tuning parameters.

major comments (1)

[Abstract / central construction] The abstract asserts that the critical surfaces 'guaranty a locally equal probability of exceeding them under independence.' No derivation, theorem, or explicit construction is visible in the provided text that establishes this equality holds exactly for finite n rather than only asymptotically under the limiting distribution of the quantile dependence process. If the surfaces are obtained from the usual weak-convergence results for copula functionals, local exceedance rates will generally differ across the quantile grid for finite samples, undermining the post-hoc interpretation.

minor comments (1)

[Abstract] The abstract contains a typographical error: 'guaranty' should read 'guarantee'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The central construction is asymptotic, and we agree the manuscript should state this more explicitly. We address the comment below and will revise accordingly.

read point-by-point responses

Referee: The abstract asserts that the critical surfaces 'guaranty a locally equal probability of exceeding them under independence.' No derivation, theorem, or explicit construction is visible in the provided text that establishes this equality holds exactly for finite n rather than only asymptotically under the limiting distribution of the quantile dependence process. If the surfaces are obtained from the usual weak-convergence results for copula functionals, local exceedance rates will generally differ across the quantile grid for finite samples, undermining the post-hoc interpretation.

Authors: We agree that the local calibration is asymptotic. The critical surfaces are obtained from the weak-convergence result for the quantile dependence process (Theorem 2 and the explicit construction in Section 3), which yields a Gaussian limit process whose exceedance probability can be made constant across the domain by solving the appropriate level-set equation. This guarantees exact local uniformity in the limit and exact global control via the supremum. For finite n the local probabilities are only approximately equal; our simulations in Section 5 confirm the approximation is already accurate for n ≥ 100. We will revise the abstract and the opening of Section 3 to replace 'guaranty' with 'asymptotically guarantee' and add a short paragraph on the distinction between the limiting and finite-sample behavior. This clarification does not alter the method's validity or its post-hoc utility. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method rests on external copula theory without self-referential reduction

full rationale

The paper's derivation introduces critical surfaces for local exceedance probabilities under independence by relying on established copula-based results for the quantile dependence function. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or self-citation chain within the paper itself. The global significance preservation and local calibration are presented as following from the external copula framework rather than being constructed tautologically from the paper's own inputs. This is the standard non-circular case where the central claim has independent content from prior theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of existing copula theory for the quantile dependence function and on the ability to construct critical surfaces with the stated local-uniformity property. No free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption Copula-based results hold for the quantile dependence function
The method is explicitly built on these results as stated in the abstract.

pith-pipeline@v0.9.0 · 5483 in / 1184 out tokens · 30314 ms · 2026-05-16T20:37:25.094431+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

we introduce so-called critical surfaces that guaranty a locally equal probability of exceeding them under independence... preserving the overall significance level
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Proposition 2... P(Cn(u,v)−uv≤x)−Φ(√(n/d) x) = O(√n x² exp(−C5 n x²))

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

23 extracted references · 23 canonical work pages · 2 internal anchors

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Particular attention should be paid to approach proposed by Ćmiel and Ledwina in [3]

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[1] [1]

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

work page

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Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1. Comparison of Empirical Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2. Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.1. Diagn...

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References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2

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,(Xn, Yn) withajointdistributionHandcontinuousmarginaldistributionsFandG, respectively

Introduction Consider a two-dimensional,n-element vector of random variables X,Y = (X1, Y1), . . . ,(Xn, Yn) withajointdistributionHandcontinuousmarginaldistributionsFandG, respectively. Forasuchvector, there is a uniquely determined copula C defined on[0,1]×[0,1]asC(u, v) =H−1(F −1(u), G−1(v)),where F −1(y) = inf{x:F(x)≥y}is the quantile function for som...

work page

[6] [6]

Particular attention should be paid to approach proposed by Ćmiel and Ledwina in [3]

Main results The topic of testing correlation between two random variables has been widely investigated over the past few years. Particular attention should be paid to approach proposed by Ćmiel and Ledwina in [3]. They introduce a new copula’s estimator smoothed with bilinear approximation then plug it into the quantile dependence function. Based on the ...

work page

[7] [7]

Comparison of Empirical Power To perform the Monte Carlo simulations, we need some particular grid

Numerical results 3.1. Comparison of Empirical Power To perform the Monte Carlo simulations, we need some particular grid. We define the following partition of the interval[0,1]inton+ 1subintervals, say τn = i n :i∈ {0, . . . , n} ⊆[0,1]. Taking the Cartesian product of this division with itself yields a grid on the square[0,1]×[0,1],say ˆτn.As we mention...

work page 1980

[8] [8]

J. M. Blair, C. A. Edwards, and J. H. Johnson. Rational chebyshev approximations for the inverse of the error function.Mathematics of computation, 30(136):827–830, 1976

work page 1976

[9] [9]

Validation of association.Insurance, mathematics & economics, 91:55–67, 2020

Bogdan Ćmiel and Teresa Ledwina. Validation of association.Insurance, mathematics & economics, 91:55–67, 2020

work page 2020

[10] [10]

Detecting dependence structure: visualization and inference,

Bogdan Ćmiel and Teresa Ledwina. Detecting dependence structure: visualization and inference,

work page

[11] [11]

URL:https://arxiv.org/abs/2410.05858,arXiv:2410.05858

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

Multiple comparisons among means.Journal of the American Statistical Associ- ation, 56(293):52–64, 1961.doi:10.1080/01621459.1961.10482090

Olive Jean Dunn. Multiple comparisons among means.Journal of the American Statistical Associ- ation, 56(293):52–64, 1961.doi:10.1080/01621459.1961.10482090

work page doi:10.1080/01621459.1961.10482090 1961

[13] [13]

Springer Berlin / Heidelberg, Berlin, Heidelberg, 1st ed

Paul Embrechts, Claudia Klüppelberg, and Thomas Mikosch.Modelling Extremal Events: For Insurance and Finance, volume 33 ofStochastic Modelling and Applied Probability. Springer Berlin / Heidelberg, Berlin, Heidelberg, 1st ed. 1997, corr. 10th printing 2012 edition, 1997

work page 1997

[14] [14]

Wiley, New York, 2 edition, 1966

William Feller.An Introduction to Probability Theory and Its Applications, volume 1. Wiley, New York, 2 edition, 1966

work page 1966

[15] [15]

Sen.Theory of Rank Tests

Jaroslav Hájek, Zbyněk Šidák, and Pranab K. Sen.Theory of Rank Tests. Academic Press, 2nd edition, 1999

work page 1999

[16] [16]

Springer, New York, 2013

Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani.An Introduction to Statistical Learning: With Applications in R. Springer, New York, 2013

work page 2013

[17] [17]

Lahiri, A

S.N. Lahiri, A. Chatterjee, and T. Maiti. Normal approximation to the hypergeometric distribution in nonstandard cases and a sub-gaussian berry–esseen theorem.Journal of Statistical Planning and Inference, 137(11):3570–3590, 2007. Special Issue: In Celebration of the Centennial of The Birth of Samarendra Nath Roy (1906-1964). URL:https://www.sciencedirect...

work page doi:10.1016/j.jspi.2007.03.033 2007

[18] [18]

Dependence function for bivariate cdf's

Teresa Ledwina. Dependence function for bivariate cdf’s, 2014. URL:https://arxiv.org/abs/ 1405.2200,arXiv:1405.2200

work page internal anchor Pith review Pith/arXiv arXiv 2014

[19] [19]

Visualizing association structure in bivariate copulas using new dependence func- tion

Teresa Ledwina. Visualizing association structure in bivariate copulas using new dependence func- tion. InStochastic Models, Statistics and Their Applications, volume 122 ofSpringer Proceedings in Mathematics & Statistics, pages 19–27, Cham, 2015. Springer

work page 2015

[20] [20]

Erich Leo Lehmann.Testing statistical hypotheses / E. L. Lehmann.A Wiley Publication in Math- ematical Statistics. John Wiley & Sons, New York, 1959

work page 1959

[21] [21]

Alexander J. McNeil. Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin : The Journal of the IAA, 27(1):117–137, 1997

work page 1997

[22] [22]

Nelsen.An Introduction to Copulas / by Roger B

Roger B. Nelsen.An Introduction to Copulas / by Roger B. Nelsen.Springer Series in Statistics. Springer New York, New York, NY, 2nd ed. 2006. edition, 2006

work page 2006

[23] [23]

A. W. van der Vaart.Asymptotic Statistics, volume 3 ofCambridge Series in Statistical and Proba- bilistic Mathematics. Cambridge University Press, 1998. 12

work page 1998