Debiased Whittle likelihood for time series and spatial data

Time series and spatial data are ubiquitous in many application areas, such as environmental data, geosciences, astronomy, and finance. A key statistical modelling and estimation challenge for these data is that of dependance between points at different times or locations. While parametric models of covariance can be estimated via exact likelihood, this is ill-suited for many practical problems due to the heavy computational cost.

A standard approach to address this relies on approximate likelihood methods. The Whittle likelihood is one such approximation for gridded data, based on the Discrete Fourier Transform of the data. It is popular due to its n log n computational cost, robustness to non-Gaussian data, and amenability to interpretation in the spectral domain. However, Whittle likelihood estimates suffer from a strong bias due to the finite and discrete sampling. This is true in particular for spatial data where bias dominates verses standard deviation in dimension equal or greater than two. Additionally, practical sampling patterns often diverge from theoretical requirements, due to non-square observational domains or missing data. In this presentation we present a recently proposed modification to the Whittle likelihood which addresses all these issues at once.
We provide asymptotic results under a framework which we call Significant Correlation Contribution, which allows us to understand the interplay between the sampling pattern and the covariance model. We demonstrate that our modification renders our estimate asymptotically efficient and normal for a wide class of settings.