Open Access
November 2020 A general frequency domain method for assessing spatial covariance structures
Matthew Van Hala, Soutir Bandyopadhyay, Soumendra N. Lahiri, Daniel J. Nordman
Bernoulli 26(4): 2463-2487 (November 2020). DOI: 10.3150/19-BEJ1160


When examining dependence in spatial data, it can be helpful to formally assess spatial covariance structures that may not be parametrically specified or fully model-based. That is, one may wish to test for general features regarding spatial covariance without presupposing any particular, or potentially restrictive, assumptions about the joint data distribution. Current methods for testing spatial covariance are often intended for specialized inference scenarios, usually with spatial lattice data. We propose instead a general method for estimation and testing of spatial covariance structure, which is valid for a variety of inference problems (including nonparametric hypotheses) and applies to a large class of spatial sampling designs with irregular data locations. In this setting, spatial statistics have limiting distributions with complex standard errors depending on the intensity of spatial sampling, the distribution of sampling locations, and the process dependence. The proposed method has the advantage of providing valid inference in the frequency domain without estimation of such standard errors, which are often intractable, and without particular distributional assumptions about the data (e.g., Gaussianity). To illustrate, we develop the method for formally testing isotropy and separability in spatial covariance and consider confidence regions for spatial parameters in variogram model fitting. A broad result is also presented to justify the method for application to other potential problems and general scenarios with testing spatial covariance. The approach uses spatial test statistics, based on an extended version of empirical likelihood, having simple chi-square limits for calibrating tests. We demonstrate the proposed method through several numerical studies.


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Matthew Van Hala. Soutir Bandyopadhyay. Soumendra N. Lahiri. Daniel J. Nordman. "A general frequency domain method for assessing spatial covariance structures." Bernoulli 26 (4) 2463 - 2487, November 2020.


Received: 1 December 2018; Revised: 1 June 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256149
MathSciNet: MR4140518
Digital Object Identifier: 10.3150/19-BEJ1160

Keywords: Confidence sets , spatial periodogram , spatial testing , spectral moment conditions , stochastic sampling

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 4 • November 2020
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