Abstract
Estimating the covariance matrix of a random vector is essential and challenging in large dimension and small sample size scenarios. The purpose of this paper is to produce an outperformed large-dimensional covariance matrix estimator in the complex domain via the linear shrinkage regularization. Firstly, we develop a necessary moment property of the complex Wishart distribution. Secondly, by minimizing the mean squared error between the real covariance matrix and its shrinkage estimator, we obtain the optimal shrinkage intensity in a closed form for the spherical target matrix under the complex Gaussian distribution. Thirdly, we propose a newly available shrinkage estimator by unbiasedly estimating the unknown scalars involved in the optimal shrinkage intensity. Both the numerical simulations and an example application to array signal processing reveal that the proposed covariance matrix estimator performs well in large dimension and small sample size scenarios.
Highlights
Formulation and the Optimal SolutionAssume a p-dimensional random vector x ∈ Cp follows the complex Gaussian distribution CN(0, Σ), where Σ is the unknown covariance matrix
During the last two decades, scientists have proposed many regularization strategies to generate outperformed covariance matrix estimators in large dimension scenarios [6,7,8,9,10]
We further research the linear shrinkage estimator under the complex Gaussian distribution. e target matrix is chosen as the spherical target which has been widely studied under the real number field [6, 11, 14]. e optimal tuning parameter is obtained by minimizing the mean squared error (MSE)
Summary
Assume a p-dimensional random vector x ∈ Cp follows the complex Gaussian distribution CN(0, Σ), where Σ is the unknown covariance matrix. Erefore, the optimal shrinkage intensity can be obtained through solving the following optimization problem: min w2Etr(T − S)2 − 2E[tr(T − S)(Σ − S)]w, (5). We can obtain the following moment property of the complex Gaussian distribution. Xn ∈ Cp follows the complex Gaussian distribution CN(0, Σ) and S is the sample covariance matrix; we have tr S2. For a random matrix W which follows complex Wishart distribution CW(Σ, n) with degree of freedom n, let Σ GGH; we have G− 1WG− H ∼ CW(Ip, n). Σ 1 − w∗S + w∗tr(pS)Ip. We remind that the optimal shrinkage estimator concerns with the real covariance matrix. We remind that the optimal shrinkage estimator concerns with the real covariance matrix. It provides a theoretical optimal value for evaluating the available ones
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