Estimation of the autoregressive parameters of a mixed autoregressive moving-average time series

The problem of estimating the autoregressive parameters of a mixed autoregressive moving-average (ARMA) time series (of known order) using the output data alone is treated. This problem is equivalent to the estimation of the denominator terms of the scalar transfer function of a stationary, linear discrete time system excited by an unobserved unenrrelated sequence input by employing only the observations of the scalar output. The solution of this problem solves the problem of the identification of the dynamics of a white-noise excited continuous-time linear stationary system using sampled data. The latter problem was suggested by Bartlett in 1946. The problem treated here has appeared before in the engineering literature. The earlier treatment yielded biased parameter estimates. An asymptotically unbiased estimator of the autoregressive parameters is obtained as the solution of a modified set of Yule-Walker equations. The asymptotic estimator covariance matrix behaves like a least-squares parameter estimate of an observation set with unknown error covariances. The estimators are also shown to be unbiased in the presence of additive independent observation noise of arbitrary finite correlation time. An example illustrates the performance of the estimating procedures.