Statistical inference for spatial statistics defined in the fourier domain

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 L2-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.