The Debiased Spatial Whittle likelihood
We provide a computationally and statistically efficient method for estimating the parameters of a stochastic Gaussian model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the debiased spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalise the approach to flexibly allow for significant volumes of missing data, for the usage of irregular sampling schemes including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings. We provide detailed implementation guidelines which ensure the estimation procedure can still be conducted in $\mathcal{O}(n\log n)$ operations, where $n$ is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and real world settings, and compare with state-of-the-art alternatives, demonstrating the enduring significant practical appeal of Fourier-based methods, provided they are corrected by the constructive procedures developed in this paper.