The Spatial Cram'{e}r--von Mises Test of Independence under $β$-Mixing: Asymptotic Theory and Python Implementation
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910234875789312 |
|---|---|
| author | Mandap, Marco |
| author_facet | Mandap, Marco |
| contents | 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 $β$-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 $θ> 2(2+δ)/δ$; (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 $β$-mixing sequences via Yoshihara (1976). The limit distribution is a weighted sum of correlated $χ^2_1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Spatial Cram'{e}r--von Mises Test of Independence under $β$-Mixing: Asymptotic Theory and Python Implementation Mandap, Marco Methodology Statistics Theory 62G10 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 $β$-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 $θ> 2(2+δ)/δ$; (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 $β$-mixing sequences via Yoshihara (1976). The limit distribution is a weighted sum of correlated $χ^2_1$ 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. |
| title | The Spatial Cram'{e}r--von Mises Test of Independence under $β$-Mixing: Asymptotic Theory and Python Implementation |
| topic | Methodology Statistics Theory 62G10 |
| url | https://arxiv.org/abs/2605.19164 |