Statistical inference for spatial statistics defined in the Fourier domain
The Annals of Statistics · 2018 · Volume 46 · Issue 2
Abstract
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 nonnegative 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 of 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.
Keywords
62M15, 62M30, Fixed and increasing domain asymptotics, irregular spaced locations, Quadratic forms, spatial spectral density function, stationary spatial random fields
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