A Central Limit Theorem for Parameter Estimation in Stationary Vector Time Series and its Application to Models for a Signal Observed with Noise

Abstract

A general finite parameter model for stationary ergodic nondeterministic vector time series is considered. A central limit theorem for parameter estimates, obtained by maximising frequency domain approximations to the Gaussian likelihood, is established. The treatment given extends the central limit theorem of Dunsmuir and Hannan in that the innovations covariance matrix and the linear transfer function need not be separately parameterised. Models for a stationary vector signal observed with stationary vector noise are discussed in relation to the central limit theorem and the conditions imposed for this result are related to this model. Finally, the special case of a scalar autoregressive signal observed with noise is discussed. It is shown that this model may be reparameterised so that the central limit theorem of Dunsmuir and Hannan may be applied.