Spectrum estimation by interpolation of covariances and cepstrum parameters in an exponentional class of spectral densities

Given output data of a stationary stochastic process estimates of the covariances and cepstrum parameters can be obtained. Methods of moments have been applied to these parameters for designing ARMA processes, and it has been shown that these two sets of parameters in fact form local coordinates for the set of ARMA processes, but that some combinations of cepstrum parameters and covariances cannot be matched exactly within this class of processes. Therefore, another class of processes is considered in this paper in order to be able to match any combination of covariances and cepstrum parameters. The main result is that a process with spectral density of the form Phi(z) = (exp{Sigma _{k = 0} ^mp_k(z^k + z^-k)})/(Sigma_{k = 0} ⁿq_k( z ^k + z^-k)/2) can always match given covariances and cepstrum parameters. This is proven using a fixed-point argument, and a non-linear least-squares problem is proposed for determining a solution