Optimal correction of an indefinite estimated MA spectral density matrix

Consider a vector moving-average sequence of order n  , MA(n)MA(n), and let Φ(ω)=∑k=-nnRke-jωk denote its spectral density matrix, where {Rk}k=-nn are the covariance matrices and ωω stands for the frequency variable. A nonparametric estimate Φ^(ω)=∑k=-nnR^ke-jωk of Φ(ω)Φ(ω) 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 Φ(ω)Φ(ω) to Φ^(ω) in a polynomial time, by means of a semidefinite programming (SDP) algorithm.