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Detecting Distributional Differences in Spatially Correlated Multivariate Data via Kernel-Smoothed Rank-Based Empirical Copula Tests

Marco Mandap

Abstract

Comparing multivariate yield quality distributions across spatially referenced agricultural fields is complicated by two pervasive features: non-normality and spatial autocorrelation. Classical procedures such as ANOVA, MANOVA, and standard rank tests assume independence and therefore exhibit severe Type I error inflation when spatial dependence is present. We propose a nonparametric spatial Cramer-von Mises-type test based on kernel-smoothed empirical copula processes constructed from pooled componentwise ranks. Spatial kernel weights account explicitly for local dependence, while the rank transformation removes sensitivity to marginal distributional form. Under fixed-domain infill asymptotics and polynomial alpha-mixing conditions, we establish weak convergence of the smoothed empirical copula process to a mean-zero Gaussian limit and show that the resulting quadratic test statistic converges to a weighted sum of chi-squared random variables restricted to the K-1-dimensional contrast subspace. Practical inference is obtained through a Satterthwaite approximation calibrated using the exact discrete spatial covariance operator under a Gaussian copula model. Monte Carlo experiments with bivariate log-normal spatial data demonstrate that the proposed test maintains nominal size across varying strengths of spatial dependence, in contrast to classical parametric and non-spatial rank-based methods, which become severely anti-conservative. The procedure provides a theoretically justified and computationally tractable framework for comparing multivariate spatial yield distributions in precision agriculture and related applied settings.

Detecting Distributional Differences in Spatially Correlated Multivariate Data via Kernel-Smoothed Rank-Based Empirical Copula Tests

Abstract

Comparing multivariate yield quality distributions across spatially referenced agricultural fields is complicated by two pervasive features: non-normality and spatial autocorrelation. Classical procedures such as ANOVA, MANOVA, and standard rank tests assume independence and therefore exhibit severe Type I error inflation when spatial dependence is present. We propose a nonparametric spatial Cramer-von Mises-type test based on kernel-smoothed empirical copula processes constructed from pooled componentwise ranks. Spatial kernel weights account explicitly for local dependence, while the rank transformation removes sensitivity to marginal distributional form. Under fixed-domain infill asymptotics and polynomial alpha-mixing conditions, we establish weak convergence of the smoothed empirical copula process to a mean-zero Gaussian limit and show that the resulting quadratic test statistic converges to a weighted sum of chi-squared random variables restricted to the K-1-dimensional contrast subspace. Practical inference is obtained through a Satterthwaite approximation calibrated using the exact discrete spatial covariance operator under a Gaussian copula model. Monte Carlo experiments with bivariate log-normal spatial data demonstrate that the proposed test maintains nominal size across varying strengths of spatial dependence, in contrast to classical parametric and non-spatial rank-based methods, which become severely anti-conservative. The procedure provides a theoretically justified and computationally tractable framework for comparing multivariate spatial yield distributions in precision agriculture and related applied settings.
Paper Structure (19 sections, 3 theorems, 15 equations, 4 tables)

This paper contains 19 sections, 3 theorems, 15 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Challenges of Spatial Yield Data
  3. Limitations of Classical Tests Under Spatial Dependence
  4. A Rank-Based Spatial Testing Framework
  5. Scope and Contributions
  6. Methodology
  7. Statistical Model and Notation
  8. Kernel-Smoothed Empirical Distribution Function
  9. Contrast Processes and Test Statistic
  10. Practical Inference via the Satterthwaite Approximation
  11. Multivariate Extension
  12. Mathematical Framework and Asymptotic Theory
  13. Simulation Study
  14. Design
  15. Univariate Results
  16. ...and 4 more sections

Key Result

Lemma 1

Assume $H_0$ so that $F_k \equiv F$ and assume the field $\{X_{k,s}\}$ is strictly stationary and $\alpha$-mixing with $\alpha(r) \le Cr^{-\theta}$ for some $\theta > 2(2+\delta)/\delta$ and some $\delta > 0$. For fixed $x, y \in \mathbb{R}$, where The integral converges absolutely.

Theorems & Definitions (3)

  • Lemma 1: Asymptotic covariance under fixed-bandwidth infill
  • Theorem 1: Weak convergence of $\tilde{\mathbb{G}}_{n,k}$
  • Theorem 2: Asymptotic null distribution