A direct derivation of the exact Fisther information matrix of Gaussian vector state space models

This paper deals with a direct derivation of Fisher's information matrix of vector state space models for the general case, by which is meant the establishment of the matrix as a whole and not element by element. The method to be used is matrix differentiation, see [4]. We assume the model to be Gaussian and use the negative logarithm of the likelihood function as used in the definition of Fisher's information. In a related paper Klein et al. [3] establish the information matrix by assembling its elements as derived in the literature [2,5,6] and for an approximation of the Hessian of the log-likelihood function one can refer to [7,8].