Bayesian-like autoregressive spectrum estimation in the case of unknown process order

Initially, the problem of estimation of the spectral density function of a stationary multivariate autoregressive Gaussian process of unknown order is considered. Two new solutions to this problem are presented. The proposed estimators, suggested by the form of the Bayesian predictor in autoregressive systems, can be characterized as the average model spectrum and the spectrum corresponding to the "averaged model," with the averages being computed over the set of competitive autoregressive models of different orders and with respect to the sequence of the posterior probabilities of the process order given its observation history. The obtained results are then extended to the case of nonstationary autoregressive processes (identified by means of the exponentially weighted estimators) and more general weighting sequences. Although not Bayesian in the strict sense, the proposed approaches seem to be interesting from the theoretical point of view and give better results than the "classical" one. The efficient computational algorithms are indicated and the results of computer simulations are discussed.