The De-Biased Whittle Likelihood

TL;DR

This paper introduces a de-biasing method for the Whittle likelihood, significantly improving parameter estimation accuracy for stationary stochastic processes while maintaining computational efficiency.

Contribution

It proposes a novel de-biasing approach for the Whittle likelihood that achieves near maximum likelihood accuracy with lower bias and retains computational efficiency.

Findings

Reduces bias by up to two orders of magnitude in simulations.

Achieves estimates close to maximum likelihood in oceanographic data.

Maintains $ ext{O}(n ext{log} n)$ computational complexity.

Abstract

The Whittle likelihood is a widely used and computationally efficient pseudo-likelihood. However, it is known to produce biased parameter estimates for large classes of models. We propose a method for de-biasing Whittle estimates for second-order stationary stochastic processes. The de-biased Whittle likelihood can be computed in the same $\mathcal{O}(n\log n)$ operations as the standard approach. We demonstrate the superior performance of the method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the de-biased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of $n^{-1/2}$, under weaker assumptions than standard theory, where we do not require that the power spectral density is continuous in frequency. We describe how the method can be easily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.