On Stein’s equation, Vandermonde matrices and Fisher’s information matrix of time series processes. Part II: The ARMAX process

An approach is presented to get interconnections between the Fisher information matrix of an ARMAX process and a corresponding solution of a Stein equation. The case of algebraic multiplicity greater than one and the case of distinct eigenvalues are addressed. Appropriate links are constructed for these two cases by applying a factorization both for the Fisher information matrix and for a corresponding solution of a Stein equation. These factored forms are nonsquare linear systems of equations Ax = b, the kernels of the appropriate coefficient matrices are described. These are of fundamental importance for the solutions of the obtained linear systems. The structured coefficient matrix associated with the factored form of the Fisher information matrix is composed by basis vectors associated with an ARMAX polynomial, whereas the coefficient matrix obtained through the solution of a Stein equation consists of resolvant matrices associated with a companion matrix used in a corresponding Stein equation. The presence of Vandermonde matrices in right inverses of coefficient matrices of the obtained linear systems is investigated. Links between coefficient matrices which originate both from the Fisher information matrix and a corresponding solution of the Stein equation are derived. An example is provided for illustrating a solution of a Stein equation in terms of the Fisher information matrix as well as for describing the kernels of the appropriate coefficient matrices. AMS classification: 15A06