| In the era of big data,the dimension and number of data have explosive growth,and the rapid increase of data dimension makes many classical matrix methods no longer applicable.In the case of large samples,the sample covariance matrix is the consistent estimator of the population covariance matrix,but in the case of high dimensions,the sample covariance matrix becomes a biased estimator.In this paper,the classical linear contraction estimation method is used to improve the deviation between the sample covariance and the population covariance matrix.This paper focuses on the selection of an optimal target matrix when Toeplitz structure information is known.Firstly,this paper takes the Ⅰ-Toeplitz type structure matrix and II-Toeplitz type structure matrix as the target matrices and puts them into the linear shrinkage estimation model to obtain the corresponding optimal shrinkage coefficient expression,and then presents a consistent estimation of the statistics with the population covariance matrix.The classical linear shrinkage estimation algorithm for Toeplitz structure matrix is summarized.Secondly,the classical linear shrinkage estimation algorithm is optimized,and the matrix variable function is summarized by selecting the statistical combinations with the same structure.The optimal target matrix is selected from the Ⅰ and Ⅱ-Toeplitz structure matrices by means of the matrix variable function values,and the optimal covariance matrix selection algorithm is proposed under the Toeplitz structure information.Then,banding method is used to improve the non-positive problem of Ⅰ-type Toeplitz structure matrix.Then,the selection algorithm of the population covariance matrix estimator under the adaptive Toeplitz structure is proposed.Finally,the consistency of the second moment in complex Gaussian distribution is verified by simulation experiment,and the validity of the algorithm is calculated by Monte Carlo simulation experiment and empirical analysis of financial data. |
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