Estimation and deconvolution when the transfer function has zeros

The problem of estimation of the transfer function and deconvolution of a linear process is considered. This paper specifically deals with the case when the transfer function has zeros on the unit circle or equivalently the spectral density function has zeros. It is shown that if the zeros are finitely many and are of finite order then we can still consistently estimate the transfer function without the minimum phase assumption when the process is non-Gaussian. Statistical properties of the estimate are given. Convergence of the deconvolution is also given. It is shown that if the transfer function vanishes on an interval, then, essentially, we cannot identify the transfer function. Two simple simulated examples are given to illustrate the procedures.