The Statistical Theory of Linear Systems

Publisher Summary The chapter discusses the development of a rather complete inferential theory for ARMAX models. The first problem in the development is the coordinatization of spaces of such structures. Coordinates are needed both for computations and because a central limit theorem must be expressed in terms of them. One such coordinatization would follow from the use of the coefficient matrices in g, h, j in the scalar m.f.d., but there are many others. The chapter highlights the algebraic and topological description of ARMAX systems. In this connection, control engineers have played a premier part. The chapter focuses on the asymptotic properties of maximum likelihood (ML) estimators. The ML estimator is obtained by optimizing this likelihood. The chapter explains the asymptotic properties of these estimators without assuming the data to be Gaussian and also discusses the basis of the assumptions that appear to be minimal.