Balancing the edge effect and dimension of spectral spatial statistics under irregular sampling with applications to isotropy testing

We investigate distributional properties of a class of spectral spatial statistics under irregular sampling of a random field that is defined on $\mathbb{R}^d$, and use this to obtain a test for isotropy. Within this context, edge effects are well-known to create a bias in classical estimators commonly encountered in the analysis of spatial data. This bias increases with dimension $d$ and, for $d>1$, can become non-negligible in the limiting distribution of such statistics to the extent that a nondegenerate distribution does not exist. We provide a general theory for a class of (integrated) spectral statistics that enables to 1) significantly reduce this bias and 2) that ensures that asymptotically Gaussian limits can be derived for $d \le 3$ for appropriately tapered versions of such statistics. We use this to address some crucial gaps in the literature, and demonstrate that tapering with a sufficiently smooth function is necessary to achieve such results. Our findings specifically shed a new light on a recent result in Subba Rao (2018a). Our theory then is used to propose a novel test for isotropy. In contrast to most of the literature, which validates this assumption on a finite number of spatial locations (or a finite number of Fourier frequencies), we develop a test for isotropy on the full spatial domain by means of its characterization in the frequency domain. More precisely, we derive an explicit expression for the minimum $L^2$-distance between the spectral density of the random field and its best approximation by a spectral density of an isotropic process. We prove asymptotic normality of an estimator of this quantity in the mixed increasing domain framework and use this result to derive an asymptotic level $\alpha$-test.