Computation of the exact information matrix of Gaussian time series with stationary random components

The paper presents an algorithm for efficient recursive computation of the Fisher information matrix of Gaussian time series whose random components are stationary, and whose means and covariances are functions of a parameter vector. The algorithm is first developed in a general framework and then specialized to the case of autoregressive moving-average processes, with possible additive white noise. The asymptotic behavior of the algorithm is explored and a termination criterion is derived. Finally, the algorithm is used to demonstrate the behavior of the exact Cramer-Rao bound (for unbiased estimates) for some ARMA processes, as a function of the number of data points. It is shown that for processes with zeros near the unit circle and short data records, the exact Cramer-Rao bound differs dramatically from its common approximation based on asymptotic theory.