An Orthogonally Equivariant Estimator of the Covariance Matrix in High Dimensions and Small Sample Size
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An orthogonally equivariant estimator for the covariance matrix is proposed that is valid when the dimension p is larger than the sample size n. Equivariance under orthogonal transformations is a less restrictive assumption than structural assumptions on the true covariance matrix. It reduces the problem of estimation of the covariance matrix to that of estimation of its eigenvalues. In this paper, the eigenvalue estimates are obtained from an adjusted likelihood function derived by approximating the integral over the eigenvectors of the sample covariance matrix, which is a challenging problem in its own right. Comparisons with two well-known orthogonally equivariant estimators are given, which are based on Monte-Carlo risk estimates for simulated data and misclassification errors in a real data analysis.
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