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.

翻译：我们研究了定义在$\mathbb{R}^d$上的随机场在不规则采样下一类谱空间统计量的分布性质，并据此构建了各向同性检验。在该背景下，边缘效应众所周知会在空间数据分析中产生经典估计量的偏差。这种偏差随维度$d$增加而增大，当$d>1$时，它可能在统计量极限分布中变得不可忽略，甚至导致非退化分布不存在。我们为（积分）谱统计量建立了一般理论，该理论能够：1）显著降低此类偏差；2）确保对于适当锥化版本的统计量，在$d \le 3$时可推导出渐近高斯极限。我们利用该理论填补了文献中的若干关键空白，并证明了使用足够光滑的锥化函数是实现这一结果的必要条件。我们的发现特别为Subba Rao（2018a）的最新结果提供了新视角。随后，基于该理论我们提出了新颖的各向同性检验。与主流文献通常通过有限空间点位（或有限傅里叶频率）验证各向同性假设不同，我们利用频域表征在全空间域上构建了各向同性检验。具体而言，我们推导了随机场谱密度与其各向同性过程谱密度最佳逼近之间最小$L^2$距离的显式表达式。在混合递增域框架下，我们证明了该估计量的渐近正态性，并据此导出了渐近水平$\alpha$-检验。