On the solution of Stein's equation and Fisher's information matrix of an ARMAX process

The main goal of this paper consists in expressing the solution of a Stein equation in terms of the Fisher information matrix (FIM) of a scalar ARMAX process. A condition for expressing the FIM in terms of a solution to a Stein equation is also set forth. Such interconnections can be derived when a companion matrix with eigenvalues equal to the roots of an appropriate polynomial associated with the ARMAX process is inserted in the Stein equation. The case of algebraic multiplicity greater than or equal to one is studied. The FIM and the corresponding solution to Stein's equation are presented as solutions to systems of linear equations. The interconnections are obtained by using the common particular solution of these systems. The kernels of the structured coefficient matrices are described as well as some right inverses. This enables us to find a solution to the newly obtained linear system of equations.