ARMA models, prewhitening, and minimum cross entropy

The problem of spectral estimation on the basis of observations from a finite stretch of a stationary time series is considered, in connection with knowledge of a prior estimate of the spectral density. A reasonable posterior spectral density estimate would be the density that is closest to the prior according to some measure of divergence, while at the same time being compatible with the data. The cross entropy has often been proposed to serve as such a measure of divergence. A correction of the original minimum-cross-entropy spectral analysis (MCESA) method of J.E. Shore (see IEEE Trans. Acoust. Speech Signal Process, vol.29, p.230-7, 1981) to traditional prewhitening techniques and to autoregressive moving average (ARMA) models is pointed out and a fast approximate solution of the minimum cross entropy problem is proposed. The solution is in a standard multiplicative form, that is, the posterior is equal to the prior multiplied by a correction factor