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

TL;DR

This paper introduces a nonparametric spatial test for comparing multivariate distributions in spatial data, effectively accounting for spatial dependence and non-normality, with proven theoretical properties and practical advantages over classical methods.

Contribution

It develops a kernel-smoothed empirical copula-based test that explicitly models spatial dependence and removes marginal distribution sensitivity, with proven asymptotic properties and practical inference procedures.

Findings

Maintains nominal size across various spatial dependence levels.

Outperforms classical methods in controlling Type I error.

Provides a computationally feasible framework for spatial distribution comparison.

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.