Empirical characteristic function identification of linear stochastic systems with possibly unstable zeros

The purpose of this paper is to adapt the empirical characteristic function (ECF) method to stable, but possibly not inverse stable linear stochastic system driven by the increments of a Lévy-process. A remarkable property of the ECF method for i.i.d. data is that, under an ideal setting, it gives an efficient estimate of the unknown parameters of a given parametric family of distributions. Variants of the ECF method for special classes of dependent data has been suggested in several papers using the joint characteristic function of blocks of unprocessed data. However, the latter may be unavailable for Lévy-systems. We introduce a new, computable score that is essentially a kind of output error. The feasibility of the procedure is based on a result of Devroye on the generation of r.v.-s with given c.f. Two special cases are considered in detail, and the asymptotic covariance matrices of the estimators are given. The present work extends our previous work on the ECF identification of stable and inverse stable linear stochastic Lévy-systems, see [1].