Generalized linear cepstral models for the spectrum of a time series

Tommaso Proietti, Alessandra Luati

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This paper introduces a class of generalized linear models with a Box–Cox link for the spectrum of a time series. The Box–Cox transformation of the spectral density is represented as a finite Fourier polynomial. Here, the coefficients of the polynomial, called generalized cepstral coefficients, provide a complete characterization of the properties of the random process. The link function depends on a power-transformation parameter, and can be expressed as an exponential model (logarithmic link), an autoregressive model (inverse link), or a moving average model (identity link). An advantage of this model class is the possibility of nesting alternative spectral estimation methods within the same likelihood-based framework. As a result, selecting a particular parametric spectrum is equivalent to estimating the transformation parameter. We also show that the generalized cepstral coefficients are a one-to-one function of the inverse partial autocorrelations of the process, which can be used to evaluate the mutual information between the past and the future of the process.

Original languageEnglish
JournalStatistica Sinica
Pages (from-to)1561-1583
Number of pages23
Publication statusPublished - 2019


  • Box–Cox link
  • Generalised linear models
  • Mutual information
  • Whittle likelihood


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