On Stein's equation, Vandermonde matrices and Fisher's information matrix of time series processes. Part I: The autoregressive moving average process

Abstract

This paper introduces several forms of relationships between Fisher's information matrix of an autoregressive-moving average or ARMA process and the solution of a corresponding Stein equation. Fisher's information matrix consists of blocks associated with the autoregressive and moving average parameters. An interconnection with a solution of Stein's equation is set forth for the block case as well as for Fisher's information matrix as a global matrix involving all parameter blocks. Both cases have their importance for the interpretation of the estimated parameters. The cases of distinct and multiple eigenvalues are addressed. The obtained links involve equations with left and right inverses, these can be expressed in terms of the inverse of appropriate Vandermonde matrices. A condition is set forth for establishing an equality between Fisher's information matrix and a solution to Stein's equation. Two examples are presented for illustrating some of the results obtained. The global and off-diagonal block case with distinct and multiple roots, respectively, are considered.