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.

翻译：提出了一种新的Toeplitz协方差矩阵非参数估计方法。该估计基于一种数据变换，将Toeplitz协方差矩阵估计问题转化为近似高斯回归中的均值估计问题。由此产生的Toeplitz协方差矩阵估计量具有正定性、完全数据驱动且计算速度极快。此外，针对一大类Toeplitz矩阵，该估计量在谱范数下被证明是极小化极大最优的。这些结果可自然推广至Toeplitz协方差矩阵逆的估计。同时，本文还基于离散余弦变换（DCT）提出了Whittle似然谱密度的替代版本。该方法已在论文配套的R包vstdct中实现。