  A new nonparametric estimator for Toeplitz covariance matrices is proposed.
This estimator is based on a data transformation that translates the problem of
Toeplitz covariance matrix estimation to the problem of mean estimation in an
approximate Gaussian regression. The resulting Toeplitz covariance matrix
estimator is positive definite by construction, fully data-driven and
computationally very fast. Moreover, this estimator is shown to be minimax
optimal under the spectral norm for a large class of Toeplitz matrices. These
results are readily extended to estimation of inverses of Toeplitz covariance
matrices. Also, an alternative version of the Whittle likelihood for the
spectral density based on the Discrete Cosine Transform (DCT) is proposed. The
method is implemented in the R package vstdct that accompanies the paper.
