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We investigate distributional properties of a class of spectral spatialstatistics under irregular sampling of a random field that is defined on$\mathbb{R}^d$, and use this to obtain a test for isotropy. Within thiscontext, edge effects are well-known to create a bias in classical estimatorscommonly encountered in the analysis of spatial data. This bias increases withdimension $d$ and, for $d>1$, can become non-negligible in the limitingdistribution of such statistics to the extent that a nondegenerate distributiondoes 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 addresssome crucial gaps in the literature, and demonstrate that tapering with asufficiently smooth function is necessary to achieve such results. Our findingsspecifically shed a new light on a recent result in Subba Rao (2018a). Ourtheory then is used to propose a novel test for isotropy. In contrast to mostof the literature, which validates this assumption on a finite number ofspatial locations (or a finite number of Fourier frequencies), we develop atest for isotropy on the full spatial domain by means of its characterizationin the frequency domain. More precisely, we derive an explicit expression forthe minimum $L^2$-distance between the spectral density of the random field andits best approximation by a spectral density of an isotropic process. We proveasymptotic normality of an estimator of this quantity in the mixed increasingdomain framework and use this result to derive an asymptotic level$\alpha$-test.

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