Generalised Linear Cepstral Models for the Spectrum of a Time Series

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.