The Debiased Spatial Whittle likelihood

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance 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 generalize the approach to flexibly allow for significant volumes of missing data 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 including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(nlogn) 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 realworld settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.

Automated summary

We propose a computationally and statistically efficient method to estimate the parameters of a stochastic covariance model observed on a regular spatial grid. Our method, called the Debiased Spatial Whittle likelihood, corrects the bias caused by boundary effects and aliasing and allows for missing data and irregular sampling boundaries. We provide a theoretical framework that ensures consistency and asymptotic normality in various practical settings, including missing data and non-Gaussian processes. The implementation guidelines ensure that the estimation procedure can be conducted in O(nlogn) operations, maintaining computational scalability for large datasets.