Statistical inference for spatial statistics defined in the Fourier domain

A class of Fourier based statistics for irregular spaced spatial data is introduced, examples include, the Whittle likelihood, a parametric estimator of the covariance function based on the $L_{2}$-contrast function and a simple nonparametric estimator of the spatial autocovariance which is a non-negative function. The Fourier based statistic is a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of the statistic is computationally tractable, requiring $O(nb^{})$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic and $n$ is the sample size. The asymptotic sampling properties of the statistic are derived using both increasing domain and fixed domain spatial asymptotics. These results are used to construct a statistic which is asymptotically pivotal.