Estimation of the autoregressive parameters from observations of a noise corrupted autoregressive time series

It has been shown that autoregressive spectral estimators can provide very fine spectral resolution estimates for time series which satisfy the all pole assumption. When the observed time series consists of the sum of an auto-regressive process plus white noise, the "all-pole" assumption is no longer valid. The appropriate model is the autoregressive-moving average representation. In this paper, it is shown that if the "higher order" Yule-Walker equations are used to estimate the autoregressive parameters of an autoregressive-moving average process, the estimates are asymptotically jointly multivariate normal. The structure of the asymptotic covariance matrix is evaluated when the process is assumed to be auto-regressive-moving average and for the special case of autoregressive plus noise.