pith. sign in

arxiv: 2605.19164 · v1 · pith:FJWXDK5Cnew · submitted 2026-05-18 · 📊 stat.ME · math.ST· stat.TH

The Spatial Cram'{e}r--von Mises Test of Independence under β-Mixing: Asymptotic Theory and Python Implementation

Marco Mandap This is my paper

Pith reviewed 2026-05-20 07:07 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords spatial statisticsCramér-von Mises testindependence testingβ-mixingU-statisticsrandom fieldsasymptotic distributionPython implementation
0
0 comments X

The pith

Under polynomial β-mixing the spatial Cramér-von Mises statistic for bivariate independence converges to a weighted sum of correlated χ² variables whose eigenvalues factor into marginal products.

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

The paper extends the classical Cramér-von Mises test of independence to stationary bivariate random fields on the plane that satisfy polynomial β-mixing dependence. It obtains the limit law by recasting the statistic as a degenerate U-statistic and invoking an extension of Gregory's theorem that applies to mixing sequences, yielding a weighted sum of χ²₁ variables whose eigenvalues are products of one-dimensional marginal eigenvalues. Explicit formulas are supplied for three weight functions, and the accompanying Python package generates Matérn fields, evaluates the statistic, and obtains critical values by Monte Carlo. A reader would care because spatial data in geography, ecology, and imaging almost always exhibit local dependence, so the usual i.i.d. critical values can be misleading.

Core claim

Under a polynomial β-mixing condition with rate θ > 2(2+δ)/δ for some δ>0, the spatial Cramér-von Mises statistic for testing independence in a stationary random field on R² converges in distribution to a weighted sum of correlated χ²₁ random variables whose eigenvalues factor as products of the marginal eigenvalues of the one-dimensional covariance operators. In the small-bandwidth limit the correlation vanishes and the limit recovers the classical independent-observations distribution. Closed-form eigenvalues are given for the uniform, optimal-normal, and Anderson-Darling weight functions, permitting direct computation of asymptotic critical values.

What carries the argument

Reformulation of the test statistic as a degenerate U-statistic of order 2 with product kernel Q = G₁ ⊗ G₂, combined with an extension of Gregory's U-statistic limit theorem to β-mixing sequences that produces the eigenvalue product structure in the limiting distribution.

Load-bearing premise

The bivariate random field is stationary and obeys a polynomial β-mixing condition with decay fast enough to make the spatial covariance kernel integrable.

What would settle it

Simulate many large samples from a Matérn field that satisfies the stated β-mixing rate and is independent across coordinates, then compare the empirical quantiles of the computed statistic against the quantiles obtained by Monte Carlo from the weighted sum of χ² variables using the paper's eigenvalue formulas.

read the original abstract

We derive the asymptotic distribution of the spatial Cram'{e}r--von Mises statistic for testing bivariate independence in stationary random fields on $\mathbb{R}^2$ under polynomial $\beta$-mixing dependence, and document the Python implementation that reproduces all simulation results. The classical test assumes i.i.d. observations; we extend it to spatially dependent data by combining three ingredients: (i) a Davydov-type covariance bound yielding integrability of the spatial covariance kernel under $\theta > 2(2+\delta)/\delta$; (ii) a reformulation of the inner-form test statistic as a degenerate U-statistic of order~2 with product kernel $Q = G_1 \otimes G_2$, following De Wet (1980); and (iii) an extension of Gregory's (1977) U-statistic limit theorem to $\beta$-mixing sequences via Yoshihara (1976). The limit distribution is a weighted sum of correlated $\chi^2_1$ variables whose eigenvalues factor as products of marginal eigenvalues; in the small-bandwidth limit the correlation vanishes and the limit reduces to the classical i.i.d. form. Explicit eigenvalue formulas are given for three weight functions (uniform, optimal normal, Anderson--Darling), producing computable critical values. The software generates Mat'{e}rn random fields by circulant embedding, computes the test statistic via the inner-form kernel decomposition, evaluates asymptotic critical values by Monte Carlo, and runs permutation-based alternatives. Simulation experiments show that the Anderson--Darling weight achieves the best power, while the Mantel and cross-$K$ tests have no power against cross-dependence in spatially correlated fields.

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

2 major / 2 minor

Summary. The paper derives the asymptotic distribution of the spatial Cramér-von Mises statistic for testing bivariate independence in stationary random fields on R² under polynomial β-mixing. It combines a Davydov-type bound for kernel integrability, a reformulation of the statistic as a degenerate order-2 U-statistic with product kernel Q = G1 ⊗ G2, and an extension of Gregory (1977) via Yoshihara (1976) to obtain a limit that is a weighted sum of correlated χ² variables whose eigenvalues factor into marginal products. Explicit eigenvalue formulas are given for uniform, optimal normal, and Anderson-Darling weights; the small-bandwidth limit recovers the classical i.i.d. form. A Python implementation using circulant embedding for Matérn fields, inner-form kernel evaluation, Monte Carlo critical values, and permutation tests is documented, with simulations indicating superior power for the Anderson-Darling weight.

Significance. If the asymptotic extension is rigorously justified, the work supplies a practical, theoretically supported test for cross-dependence in spatially correlated data together with computable critical values. Notable strengths are the explicit eigenvalue expressions for three weight functions, the reproducible Python code that generates all simulation results, and the clear demonstration that Mantel and cross-K tests lose power under spatial correlation while the proposed statistic retains it.

major comments (2)
  1. [Section 3 (Asymptotic Theory) and the paragraph following Eq. (12)] The central derivation invokes Yoshihara (1976) to extend Gregory (1977) to β-mixing sequences after reformulating the statistic as a degenerate U-statistic. Because the observations are drawn from a stationary random field on R², the pairs of locations induce a two-dimensional dependence structure; the manuscript applies the one-dimensional theorem directly without an explicit reduction to a one-dimensional mixing sequence or verification that all hypotheses of Yoshihara (uniform integrability of the degenerate projection, control of remainder terms) continue to hold under the stated polynomial rate θ > 2(2+δ)/δ. This justification is load-bearing for the claimed limit distribution and the subsequent eigenvalue formulas.
  2. [Lemma 2.3 and the proof sketch of Theorem 3.2] The Davydov-type bound is used to secure ∫ |Cov| dx < ∞, yet the manuscript does not confirm that this integrability plus the mixing rate automatically transfers the remaining technical conditions (e.g., moment bounds on the kernel projections) required by the U-statistic limit theorem in the spatial setting. A short appendix lemma addressing this transfer would strengthen the argument.
minor comments (2)
  1. [Introduction, paragraph 3] The abstract states the mixing rate θ > 2(2+δ)/δ but the introduction would benefit from repeating this condition with a brief explanation of its origin in the Davydov inequality.
  2. [Section 2.2] Notation for the spatial covariance kernel K(x,y) is introduced in the appendix; a one-sentence reminder of its definition in the main text before Eq. (8) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify places where the justification for the asymptotic extension can be made more explicit. We address each point below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Section 3 (Asymptotic Theory) and the paragraph following Eq. (12)] The central derivation invokes Yoshihara (1976) to extend Gregory (1977) to β-mixing sequences after reformulating the statistic as a degenerate U-statistic. Because the observations are drawn from a stationary random field on R², the pairs of locations induce a two-dimensional dependence structure; the manuscript applies the one-dimensional theorem directly without an explicit reduction to a one-dimensional mixing sequence or verification that all hypotheses of Yoshihara (uniform integrability of the degenerate projection, control of remainder terms) continue to hold under the stated polynomial rate θ > 2(2+δ)/δ. This justification is load-bearing for the claimed limit distribution and the subsequent eigenvalue formulas.

    Authors: We appreciate the referee’s observation that the direct invocation of the one-dimensional result requires an explicit bridge to the spatial setting. In the revision we will insert a short paragraph immediately after the statement of Theorem 3.2 that (i) orders the finite set of locations by a lexicographic traversal compatible with the polynomial mixing rate, thereby converting the field into a triangular array that satisfies the hypotheses of Yoshihara (1976), and (ii) verifies that the Davydov integrability already established in Lemma 2.3 supplies the uniform integrability of the degenerate projection and the control of remainder terms under θ > 2(2+δ)/δ. This makes the reduction transparent without altering the limit statement. revision: yes

  2. Referee: [Lemma 2.3 and the proof sketch of Theorem 3.2] The Davydov-type bound is used to secure ∫ |Cov| dx < ∞, yet the manuscript does not confirm that this integrability plus the mixing rate automatically transfers the remaining technical conditions (e.g., moment bounds on the kernel projections) required by the U-statistic limit theorem in the spatial setting. A short appendix lemma addressing this transfer would strengthen the argument.

    Authors: We agree that an explicit transfer lemma would strengthen the argument. We have added a concise Appendix Lemma A.1 that shows how the covariance integrability obtained from the Davydov bound, together with the polynomial β-mixing rate, implies the moment bounds on the first-order projections of the product kernel Q = G₁ ⊗ G₂ and the uniform integrability required by the spatial U-statistic limit theorem. The proof of the new lemma uses only standard mixing inequalities and does not rely on additional assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external theorems to spatial setting

full rationale

The paper obtains the limiting distribution of the spatial CvM statistic by (i) a Davydov-type bound for integrability of the covariance kernel under the stated polynomial β-mixing rate, (ii) reformulation as a degenerate U-statistic of order 2 with product kernel following the external reference De Wet (1980), and (iii) application of an extension of Gregory (1977) via the external reference Yoshihara (1976) for β-mixing sequences. These steps cite independent prior results whose authors do not overlap with the present paper. No equation or claim reduces the final weighted-sum-of-χ² limit (or its eigenvalue factorization) to a quantity that is fitted, defined, or presupposed inside the current manuscript. The derivation therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard mixing and U-statistic theory plus one domain-specific rate condition; no new entities or fitted parameters are introduced.

axioms (2)
  • domain assumption Polynomial β-mixing with θ > 2(2+δ)/δ for some δ>0
    Invoked to guarantee integrability of the spatial covariance kernel via Davydov-type bound.
  • standard math Extension of Gregory’s U-statistic limit theorem to β-mixing sequences via Yoshihara (1976)
    Used to obtain the weighted sum of correlated χ² variables.

pith-pipeline@v0.9.0 · 5844 in / 1355 out tokens · 51421 ms · 2026-05-20T07:07:42.904621+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

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

  1. [1]

    Equivalent comparisons of experiments.The Annals of Mathematical Statis- tics, 24(2):265–272, 1953

    Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain “good- ness of fit” criteria based on stochastic processes.Annals of Mathematical Statistics, 23(2):193–212.doi:10.1214/aoms/1177729437

    work page doi:10.1214/aoms/1177729437 1952

  2. [2]

    Billingsley,Convergence of Probability Measures

    Billingsley, P. (1999).Convergence of Probability Measures(2nd ed.). Wiley. doi:10.1002/9780470316962

    work page doi:10.1002/9780470316962 1999

  3. [3]

    Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions.Probability Surveys, 2:107–144. doi:10.1214/154957805100000104

    work page doi:10.1214/154957805100000104 2005

  4. [4]

    Davydov, Yu. A. (1970). The invariance principle for stationary processes.Theory of Probability & Its Applications, 15(3):487–498.doi:10.1137/1115050

    work page doi:10.1137/1115050 1970

  5. [5]

    and Taqqu, M

    Dehling, H. and Taqqu, M. S. (1989). The empirical process of some long-range depen- dent sequences with an application toU-statistics.Annals of Statistics, 17(4):1767– 1783.doi:10.1214/aos/1176347394

    work page doi:10.1214/aos/1176347394 1989

  6. [6]

    (1994).Mixing: Properties and Examples

    Doukhan, P. (1994).Mixing: Properties and Examples. Springer. doi:10.1007/978-1-4612-2642-0

    work page doi:10.1007/978-1-4612-2642-0 1994

  7. [7]

    Doukhan, P., Lang, G., and Surgailis, D. (2002). Asymptotics of weighted empirical processes of linear fields with long-range dependence.Annales de l’IHP Probabilit´ es et statistiques, 38(6):879–896

    work page 2002

  8. [8]

    De Wet, T. (1980). Cram´ er–von Mises tests for independence.Journal of Multivariate Analysis, 10(1):38–50.doi:10.1016/0047-259X(80)90080-9

    work page doi:10.1016/0047-259x(80)90080-9 1980

  9. [9]

    Durbin, J. (1973). Weak convergence of the sample distribution func- tion when parameters are estimated.Annals of Statistics, 1(2):279–290. doi:10.1214/aos/1176342365

    work page doi:10.1214/aos/1176342365 1973

  10. [10]

    Gregory, G. G. (1977). Large sample theory forU-statistics and tests of fit.Annals of Statistics, 5(1):110–123.doi:10.1214/aos/1176343744

    work page doi:10.1214/aos/1176343744 1977

  11. [11]

    Ibragimov, I. A. (1962). Some limit theorems for stationary processes.Theory of Probability & Its Applications, 7(4):349–382.doi:10.1137/1107036

    work page doi:10.1137/1107036 1962

  12. [12]

    Lahiri, S. N. (2003).Resampling Methods for Dependent Data. Springer. doi:10.1007/978-1-4757-3487-5

    work page doi:10.1007/978-1-4757-3487-5 2003

  13. [13]

    Lee, A. J. (1990).U-Statistics: Theory and Practice. Marcel Dekker

    work page 1990

  14. [14]

    and Neumann, M

    Leucht, A. and Neumann, M. H. (2013). Dependent wild bootstrap for degenerate U- andV-statistics.Journal of Statistical Planning and Inference, 143(8):1365–1384. doi:10.1016/j.jspi.2013.02.005

    work page doi:10.1016/j.jspi.2013.02.005 2013

  15. [15]

    Mantel, N. (1967). The detection of disease clustering and a generalized regression approach.Cancer Research, 27(2):209–220

    work page 1967

  16. [16]

    Ripley, B. D. (1976). The second-order analysis of stationary point processes.Journal of Applied Probability, 13(2):255–266. 33

    work page 1976

  17. [17]

    Serfling, R. J. (1980).Approximation Theorems of Mathematical Statistics. Wiley

    work page 1980

  18. [18]

    J., Rizzo, M

    Sz´ ekely, G. J., Rizzo, M. L., and Bakirov, N. K. (2007). Measuring and test- ing dependence by correlation of distances.Annals of Statistics, 35(6):2769–2793. doi:10.1214/009053607000000505

    work page doi:10.1214/009053607000000505 2007

  19. [19]

    van der Vaart, A. W. and Wellner, J. A. (1996).Weak Convergence and Empirical Processes. Springer.doi:10.1007/978-1-4757-2545-2

    work page doi:10.1007/978-1-4757-2545-2 1996

  20. [20]

    Wand, M. P. and Jones, M. C. (1995).Kernel Smoothing. Chapman and Hall/CRC

    work page 1995

  21. [21]

    (2004).Scattered Data Approximation

    Wendland, H. (2004).Scattered Data Approximation. Cambridge University Press

    work page 2004

  22. [22]

    Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian pro- cesses in [0,1] d.Journal of Computational and Graphical Statistics, 3(4):409–432. doi:10.1080/10618600.1994.10474655

    work page doi:10.1080/10618600.1994.10474655 1994

  23. [23]

    Yoshihara, K. (1976). Limiting behavior ofU-statistics for stationary, absolutely regular processes.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 35:237–252.doi:10.1007/BF00532676. 34

    work page doi:10.1007/bf00532676 1976