pith. sign in

arxiv: 2505.08045 · v2 · pith:KBEKW7VXnew · submitted 2025-05-12 · 🧮 math.ST · stat.TH

Measures of association for approximating copulas

Marcus Rockel This is my paper

Pith reviewed 2026-05-25 07:49 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords copulasChatterjee's ξcheckerboard approximationassociation measuresTP2 densityBernstein copulasshuffle-of-min copulas
0
0 comments X

The pith

Checkerboard copula approximations to absolutely continuous TP2 copulas satisfy ξ(C_n) ≤ ξ(C) with convergence as n increases.

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

The paper derives closed-form expressions for Chatterjee's ξ and other association measures on Bernstein, shuffle-of-min, checkerboard, and check-min copulas. It proves an inequality and convergence result: when an absolutely continuous bivariate copula C with TP2 density is approximated by its n by n checkerboard version C_n, the dependence measure ξ on the approximation is at most the value on C and approaches it in the limit. These findings matter for users of copula approximations because they supply exact, computable values of dependence strength instead of requiring simulation and they bound the error introduced by the discretization.

Core claim

Given an absolutely continuous bivariate copula C with TP2 density and its n×n-checkerboard approximation C_n, ξ(C_n) ≤ ξ(C) and ξ(C_n) → ξ(C) as n→∞. Closed-form expressions are supplied for Chatterjee's ξ on Bernstein, shuffle-of-min, checkerboard, and check-min copulas.

What carries the argument

The n×n-checkerboard copula C_n obtained by discretizing the original copula on a uniform grid, together with the inequality it induces on Chatterjee's ξ.

If this is right

  • Exact formulas for ξ on the listed approximating copulas allow direct evaluation of dependence without numerical methods.
  • The inequality shows that checkerboard approximations supply lower bounds on the dependence strength measured by ξ.
  • Convergence guarantees that refining the grid size n recovers the original dependence measure in the limit.

Where Pith is reading between the lines

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

  • The same discretization technique and inequality might extend to other grid-based copula approximations beyond the checkerboard case.
  • These closed forms could be used to derive explicit error bounds when dependence measures are computed from approximated copulas in statistical applications.
  • The results suggest testing whether similar monotonicity holds for other popular dependence coefficients under the same TP2 and absolute-continuity conditions.

Load-bearing premise

The copula C must be absolutely continuous and possess a TP2 density.

What would settle it

An absolutely continuous TP2 copula C for which ξ(C_n) > ξ(C) holds for some finite n, or for which the sequence ξ(C_n) fails to approach ξ(C).

Figures

Figures reproduced from arXiv: 2505.08045 by Marcus Rockel.

Figure 1
Figure 1. Figure 1: The estimator ξ κ n for different values of κ. Each boxplot corresponds to an increasing sample size n. The estimates concentrate near the theoretical value of ξ (red line), illustrating consistency. Whilst in the above example, different values of κ were considered, it is also interesting to compare the performance of the estimator ξ κ n with the classical Chatterjee estimator ξn defined above in (19). In… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of xi estimates to the true value as sample size increases. As suggested by [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Execution time scaling for different estimation methods with increasing sample size. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

This paper studies closed-form expressions for multiple association measures of copulas commonly used for approximation purposes, including Bernstein, shuffle--of--min, checkerboard and check--min copulas. In particular, closed-form expressions are provided for the recently popularized Chatterjee's $\xi$, which quantifies the dependence between two random variables. Given an absolutely continuous bivariate copula $C$ with TP$_2$ density and approximating $n\times n$-checkerboard copula $C_n$, we show that $\xi(C_n) \le \xi(C)$ with $\xi(C_n) \to \xi(C)$ as $n\to\infty$.

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.

Referee Report

0 major / 0 minor

Summary. The paper derives closed-form expressions for association measures including Chatterjee's ξ for Bernstein, shuffle-of-min, checkerboard, and check-min copulas. It proves that for an absolutely continuous bivariate copula C with TP₂ density, the checkerboard approximation C_n satisfies ξ(C_n) ≤ ξ(C) and ξ(C_n) → ξ(C) as n → ∞.

Significance. The explicit, parameter-free formulas for ξ on these families are verifiable by substitution into the definition and represent a practical contribution. The monotonicity and convergence result under the TP2 density condition is a load-bearing but well-stated theorem that could support further work on approximation quality in copula dependence measures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation consists of closed-form algebraic and integral expressions for Chatterjee's ξ on Bernstein, shuffle-of-min, checkerboard and check-min copulas, together with a direct proof of the inequality ξ(C_n) ≤ ξ(C) and the limit ξ(C_n) → ξ(C) as n → ∞ under the stated hypotheses of absolute continuity and TP₂ density. These steps rest on substitution into the definition of ξ and standard copula properties; no parameter is fitted and then relabeled as a prediction, no self-citation chain is invoked to justify a uniqueness claim, and no ansatz is smuggled in. The central results are therefore self-contained and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of absolute continuity and TP2 density for the copula; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The bivariate copula C is absolutely continuous with TP2 density.
    Explicitly required in the abstract for the inequality and convergence result to hold.

pith-pipeline@v0.9.0 · 5617 in / 1278 out tokens · 44994 ms · 2026-05-25T07:49:13.056076+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

  1. [1]

    A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

    Jonathan Ansari and Sebastian Fuchs. A simple extension of azadkia and chatterjee’s rank correlation to a vector of endogenous variables.arXiv preprint arXiv:2212.01621, 2022

    work page arXiv 2022

  2. [2]

    On continuity of Chatterjee's rank correlation and related dependence measures

    Jonathan Ansari and Sebastian Fuchs. On continuity of chatterjee’s rank correlation and related dependence measures.arXiv preprint arXiv:2503.11390, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    Langthaler, Sebastian Fuchs, and Wolfgang Trutschnig

    Jonathan Ansari, Patrick B. Langthaler, Sebastian Fuchs, and Wolfgang Trutschnig. Quan- tifying and estimating dependence via sensitivity of conditional distributions.arXiv preprint arXiv:2308.06168, 2023

    work page arXiv 2023

  4. [4]

    Dependence properties of bivariate copula families

    Jonathan Ansari and Marcus Rockel. Dependence properties of bivariate copula families. Dependence Modeling, 12(1):20240002, 2024

    work page 2024

  5. [5]

    Azadkia and S

    M. Azadkia and S. Chatterjee. A simple measure of conditional dependence.Ann. Stat., 49(6):3070–3102, 2021

    work page 2021

  6. [6]

    Chatterjee

    S. Chatterjee. A new coefficient of correlation.J. Amer. Statist. Ass., 116(536):2009–2022, 2020. 20

    work page 2009

  7. [7]

    From bernstein polynomials to bernstein copulas.J

    Claudia Cottin and Dietmar Pfeifer. From bernstein polynomials to bernstein copulas.J. Appl. Funct. Anal, 9(3-4):277–288, 2014

    work page 2014

  8. [8]

    A copula-based non-parametric measure of regression dependence.Scandinavian Journal of Statistics, 40(1):21–41, 2013

    Holger Dette, Karl F Siburg, and Pavel A Stoimenov. A copula-based non-parametric measure of regression dependence.Scandinavian Journal of Statistics, 40(1):21–41, 2013

    work page 2013

  9. [9]

    A typical copula is singular.Journal of Mathematical Analysis and Applications, 430(1):517–527, 2015

    Fabrizio Durante, Juan Fernandez-Sanchez, and Wolfgang Trutschnig. A typical copula is singular.Journal of Mathematical Analysis and Applications, 430(1):517–527, 2015

    work page 2015

  10. [10]

    Boca Raton, FL: CRC Press, 2016

    Fabrizio Durante and Carlo Sempi.Principles of Copula Theory. Boca Raton, FL: CRC Press, 2016

    work page 2016

  11. [11]

    Copulas approximation and new families.Available at SSRN 1032547, 2000

    Valdo Durrleman, Ashkan Nikeghbali, and Thierry Roncalli. Copulas approximation and new families.Available at SSRN 1032547, 2000

    work page 2000

  12. [12]

    Total positivity of copulas from a markov kernel perspective.Journal of Mathematical Analysis and Applications, 518(1):126629, 2023

    Sebastian Fuchs and Marco Tschimpke. Total positivity of copulas from a markov kernel perspective.Journal of Mathematical Analysis and Applications, 518(1):126629, 2023

    work page 2023

  13. [13]

    Harry Joe.Multivariate Models and Dependence Concepts, volume 73 ofMonogr. Stat. Appl. Probab.London: Chapman and Hall, 1997

    work page 1997

  14. [14]

    Checkerboard copula defined by sums of random variables.Dependence Modeling, 8(1):70–92, 2020

    Viktor Kuzmenko, Romel Salam, and Stan Uryasev. Checkerboard copula defined by sums of random variables.Dependence Modeling, 8(1):70–92, 2020

    work page 2020

  15. [15]

    On approximation of copulas.Distributions with given marginals and moment problems, pages 107–116, 1997

    X Li, P Mikusi´ nski, H Sherwood, and MD Taylor. On approximation of copulas.Distributions with given marginals and moment problems, pages 107–116, 1997

    work page 1997

  16. [16]

    Strong approximation of copulas.Journal of Mathematical Analysis and Applications, 225(2):608–623, 1998

    Xin Li, P Mikusi´ nski, and Michael D Taylor. Strong approximation of copulas.Journal of Mathematical Analysis and Applications, 225(2):608–623, 1998

    work page 1998

  17. [17]

    Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions.Journal of Multivariate Analysis, 141:197–216, 2015

    Georg Mainik. Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions.Journal of Multivariate Analysis, 141:197–216, 2015

    work page 2015

  18. [18]

    Shuffles of min.Stochastica, 13(1):61–74, 1992

    Piotr Mikusinski, Howard Sherwood, and Michael D Taylor. Shuffles of min.Stochastica, 13(1):61–74, 1992

    work page 1992

  19. [19]

    Piotr Mikusi´ nski and Michael D. Taylor. Some approximations ofn-copulas.Metrika, 72(3):385–414, 2010

    work page 2010

  20. [20]

    Nelsen.An Introduction to Copulas

    Roger B. Nelsen.An Introduction to Copulas. 2nd ed.New York, NY: Springer, 2006

    work page 2006

  21. [21]

    New copu- las based on general partitions-of-unity and their applications to risk management.Dependence modeling, 4(1):000010151520160006, 2016

    Dietmar Pfeifer, Herv´ e Awoumlac Tsatedem, Andreas M¨ andle, and Cˆ ome Girschig. New copu- las based on general partitions-of-unity and their applications to risk management.Dependence modeling, 4(1):000010151520160006, 2016

    work page 2016

  22. [22]

    The bernstein copula and its applications to modeling and approximations of multivariate distributions.Econometric theory, 20(3):535–562, 2004

    Alessio Sancetta and Stephen Satchell. The bernstein copula and its applications to modeling and approximations of multivariate distributions.Econometric theory, 20(3):535–562, 2004

    work page 2004

  23. [23]

    On the exact region determined by kendall’sτand spearman’sρ.Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(2):613–633, 2017

    Manuela Schreyer, Roland Paulin, and Wolfgang Trutschnig. On the exact region determined by kendall’sτand spearman’sρ.Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(2):613–633, 2017

    work page 2017

  24. [24]

    On the minimum information checkerboard copulas under fixed kendall’s rank correlation.arXiv preprint arXiv:2306.01604, 2023

    Issey Sukeda and Tomonari Sei. On the minimum information checkerboard copulas under fixed kendall’s rank correlation.arXiv preprint arXiv:2306.01604, 2023

    work page arXiv 2023

  25. [25]

    Revisiting the region deter- mined by spearman’sρand spearman’s footruleϕ.Journal of Computational and Applied Mathematics, 457:116259, 2025

    Marco Tschimpke, Manuela Schreyer, and Wolfgang Trutschnig. Revisiting the region deter- mined by spearman’sρand spearman’s footruleϕ.Journal of Computational and Applied Mathematics, 457:116259, 2025

    work page 2025

  26. [26]

    Beta function.https://mathworld

    Eric W Weisstein. Beta function.https://mathworld. wolfram. com/, 2002

    work page 2002

  27. [27]

    Approximation of bivariate copulas by patched bivariate fr´ echet copulas.Insurance: Mathematics and Economics, 48(2):246–256, 2011

    Yanting Zheng, Jingping Yang, and Jianhua Z Huang. Approximation of bivariate copulas by patched bivariate fr´ echet copulas.Insurance: Mathematics and Economics, 48(2):246–256, 2011. 21

    work page 2011