Optimal correction of an indefinite estimated MA spectral density matrix

Consider a vector moving-average sequence of order n, MA(n), and let denote its spectral density matrix, where are the covariance matrices and [omega] stands for the frequency variable. A nonparametric estimate of [Phi]([omega]) can easily become indefinite at some frequencies, and thus invalid, due to the estimation errors. In this paper, we provide a computationally efficient procedure that obtains the optimal (in a least-squares sense) valid approximation [Phi]([omega]) to in a polynomial time, by means of a semidefinite programming (SDP) algorithm.