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October 2021 Reconciling the Gaussian and Whittle likelihood with an application to estimation in the frequency domain
Suhasini Subba Rao, Junho Yang
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Ann. Statist. 49(5): 2774-2802 (October 2021). DOI: 10.1214/21-AOS2055


In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussian likelihood and the Whittle likelihood. 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 time series. Such a transformation yields a new decomposition for the inverse of a Toeplitz matrix and enables the representation of the Gaussian likelihood within the frequency domain. We show that the difference between the Gaussian and Whittle likelihood is due to the omission of the best linear predictions outside the domain of observation in the periodogram associated with the Whittle likelihood. Based on this result, we obtain an approximation for the difference between the Gaussian and Whittle likelihoods in terms of the best fitting, finite order autoregressive parameters. These approximations are used to define two new frequency domain quasi-likelihood criteria. We show that these new criteria can yield a better approximation of the spectral divergence criterion, as compared to both the Gaussian and Whittle likelihoods. In simulations, we show that the proposed estimators have satisfactory finite sample properties.

Funding Statement

SSR and JY gratefully acknowledge the support of the National Science Foundation (grant DMS-1812054).
The first author was also supported in part by the NSF DMS-1513647.


The research was partially conducted while SSR was visiting the Universität Heidelberg, SSR is extremely gratefully to the hospitality of everyone at Mathematikon. SSR is grateful to Thomas Hotz for several fruitful discussions. Finally, this paper is dedicated to SSR’s Father, Tata Subba Rao, who introduced the Whittle likelihood to (the young and rather confused) SSR many years ago.

The authors wish to thank three anonymous referees and editors for their insightful observations and suggestions, which substantially improved all aspects of the paper.

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Download Citation

Suhasini Subba Rao. Junho Yang. "Reconciling the Gaussian and Whittle likelihood with an application to estimation in the frequency domain." Ann. Statist. 49 (5) 2774 - 2802, October 2021.


Received: 1 January 2020; Revised: 1 January 2021; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2055

Primary: 62M10 , 62M15
Secondary: 62F10

Keywords: Biorthogonal transforms , discrete Fourier transform , periodogram , quasi-likelihoods , second order stationary time series , Toeplitz inverse

Rights: Copyright © 2021 Institute of Mathematical Statistics