A spectral domain test of isotropic properties for irregularly spaced spatial data

In modelling spatial data, it is a crucial aspect to specify the covariance function of the random field appropriately. For the sake of simplicity, the spatial isotropy is often assumed. By approximating the isotropy by a composite hypothesis containing the rotational invariance and axial symmetry of the covariance function, a maximum statistic is proposed to test the assumption of isotropy. The proposed test statistic is constructed by maximizing two Anderson-Darling (A-D) statistics, in which one is built up based on spatial periodogram-ratios of the random field at one sampling location set and its rotated version, and the other is based on spatial periodogram-ratios at the sampling location set and its axial symmetric one. Under the null, the probability distribution of the proposed maximum statistic can be approximated by simulation. The proposed nonparametric test is independent of any smoothing parameters, and is applicable for analyzing irregularly spaced spatial data.