A generalization of Whittle's formula for the information matrix of vector mixed time series

In a pioneering paper Whittle developed a formula for expressing Fisher's information matrix of multivariate time series models (cf. P. Whittle, J. Royal Statist Soc. B 15 (1953) 125-139). It is described as a function of the spectral density of the time series process. The existing relationship is extended to the whole matrix instead of one element and is related with a time domain alternative expression. The latter derives Fisher's information matrix from the log Gaussian likelihood function. The equivalence of both approaches, frequency and time domain, which is summarized in a theorem, shows that a considerable reduction in matrix integrals is taking place when moving from the former to the latter. The Hermitian property of the matrices under study contributes to construct the link between the two approaches, and the theorem is further illustrated by an example.