# UCL Discovery

## The De-Biased Whittle Likelihood

Sykulski, AM; Olhede, SC; Guillaumin, AP; Lilly, JM; Early, JJ; The De-Biased Whittle Likelihood.

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## 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.

Type: Article The De-Biased Whittle Likelihood To appear shortly in Biometrika. Full published version includes extensions of theory to non-Gaussian processes, and new simulation examples with an AR(4) and non-Gaussian process stat.ME, stat.ME, math.ST, stat.CO, stat.ML, stat.TH UCL > Provost and Vice Provost OfficesUCL > Provost and Vice Provost Offices > UCL BEAMSUCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical SciencesUCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science http://discovery.ucl.ac.uk/id/eprint/1496575