On the estimation of the parameters of vector Gaussian processes from sample covariances

Estimation of the parameters of stationary time series from a finite set of sample covariances is known to be inefficient in general, i.e. the Cramer-Rao lower bound is not achieved, even asymptotically. This paper considers the specific case of vector Gaussian processes whose second-order moments depend on a finite number of parameters. It is shown that (under certain regularity conditions) estimates of the parameters that are computed from the sample covariances can be made asymptotically efficient, if the number of sample covariances is allowed to grow with the number of data. This result holds, as a special case, for autoregressive moving average processes whose zeros are strictly inside the unit circle.