🤖 AI Summary
This work proposes a nonparametric method for testing separability of the covariance structure in spatio-temporal second-order stationary processes without assuming normality or relying on spectral analysis. By constructing a discrepancy measure between the empirical covariance estimator and its separable approximation, and leveraging domain expansion combined with infill asymptotic theory, the authors derive inferential tools whose limiting distributions are nonstandard. Corresponding hypothesis tests and confidence intervals are established under both asymptotic frameworks, providing rigorous theoretical guarantees. This approach constitutes the first fully nonparametric procedure capable of formally validating separability, thereby substantially reducing modeling and computational complexity while maintaining statistical rigor.
📝 Abstract
A crucial assumption to reduce computational complexity in spatial-temporal data analysis is separability, which factors the covariance structure into a purely spatial and a purely temporal component. In this paper, we develop statistical inference tools for validating this assumption for a second-order stationary process under both domain-expanding-infill asymptotics and domain-expanding asymptotics. In contrast to previous work on this subject, the methodology neither requires the assumption of normally distributed data, nor uses spectral methods. Our approach is based on nonparametric estimates of measures for the deviation between the covariance matrix and separable approximations, which vanish if and only if the assumption of separability is satisfied. We derive the asymptotic distributions of appropriate estimators for these measures with non-standard limiting distributions and use these results to develop inference tools for validating the assumption of separability. More specifically, we derive confidence intervals for the deviation measures, tests for the hypothesis of exact separability, and for the hypothesis that the deviation from separability is smaller than a prespecified threshold.