### Reconciling the Gaussian and Whittle Likelihood with anapplication to estimation in the frequency domain

Statistics Seminar

9th January 2020, 3:00 pm – 3:45 pm

Fry Building, G.07

In time series analysis, there are researchers who work in the time domain and other swho work in the frequency domain. The dichotomy is almost as stark as that between Frequentists and Bayesians. The aim of this talk is to investigate the connections between frequency domain and time domain methods. Our focus will be on reconciling the Gaussian likelihood (time domain) and the Whittle likelihood (frequency domain).It is well known that the Whittle likelihood is an approximation of the Gaussian likelihood.But how they are connected has received practically no attention. Our objective is to study this relationship. It is well known that the Whittle likelihood can suffer from a substantial finite sample bias. A proper understanding of the relationship between the two likelihoods, motivates improved frequency domain estimation methods which are comparable to (or can possibly improve on) time domain methods.

A more detailed outline of the talk is given below.

In this talk we derive an exact, interpretable, bound between the Gaussian and Whittle likelihood of a second order stationary time series. The derivation is based on obtaining the transformation which is biorthogonal to the discrete Fourier transform of the timeseries. Such a transformation enables the representation of the Gaussian Likelihood within the frequency domain. We show that that the approximation error between the Gaussian and Whittle likelihood is dueto the omission of the best linear predictions outside the domain of observation in the periodogram associated with the Whittle Likelihood. For autoregressive processes of finite order we derive analytic expression for the difference between the two likelihoods.For general time series models with an infinite order autoregressive representations, we obtain an approximation for this differences in terms of the infinite order autoregressive parameters. We show that a coarse approximation can be made in terms of a finite order autoregressive process. These results motivate two new frequency domain quasi-likelihoods, which are based on predicting the observed time series outside the boundary of observations. Interestingly we show these new criterions yield a better estimator of the spectral divergence criterion, as compared with both the Gaussian and Whittlelikelihoods.In simulations we show that the proposed estimators have satisfactory finite sample properties, often outperforming some of the standard methods for parameter estimation.

*Organiser*: Song Liu

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