2026/05/18 by Marco Mandap
#stat.ME #math.ST #stat.TH
We introduce a general-dimensional origin-invariant Cramér--von Mises statistic ωd2 for testing complete spatial randomness, defined by averaging corner-oriented empirical-distribution-function discrepancies over all 2d corners of [0,1]d. The statistic has a closed-form O(n2) computing formula. For d=1 it reduces to the classical rank-based Cramér--von Mises statistic, while for d=2 its computing formula agrees with Zimmerman's origin-invariant statistic. Under iid complete spatial randomness, the associated empirical process converges in ℓ^∞ to a centered Gaussian process with an explicit cross-corner covariance kernel. The continuous mapping theorem yields a quadratic Gaussian limit governed by a positive trace-class covariance operator with trace 2-d-3-d. We also connect fixed-count complete spatial randomness with the homogeneous Poisson formulation. For strictly stationary alpha-mixing sequences with uniform marginals, we establish covariance summability, the long-run variance limit, and a finite-dimensional all-corners Gaussian limit. Monte Carlo experiments illustrate null calibration and sensitivity to selected alternatives. The statistic is consistent against fixed iid alternatives whose distribution functions differ from uniformity on a set of positive measure.