Non-parametric shrinkage mean estimation for quadratic loss functions with unknown covariance matrices

Abstract

In this paper, a shrinkage estimator for the population mean is proposed under known quadratic loss functions with unknown covariance matrices. The new estimator is non-parametric in the sense that it does not assume a specific parametric distribution for the data and it does not require the prior information on the population covariance matrix. Analytical results on the improvement of the proposed shrinkage estimator are provided and some corresponding asymptotic properties are also derived. Finally, we demonstrate the practical improvement of the proposed method over existing methods through extensive simulation studies and real data analysis. Keywords: High-dimensional data; Shrinkage estimator; Large $p$ small $n$; $U$-statistic.

Some technical parts of Theorem 3.1 and 3.2 were corrected in this version

References in corpus (5)

• Two sample tests for high-dimensional covariance matrices
• Sparse linear discriminant analysis by thresholding for high dimensional data
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• High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation
• Improved multivariate normal mean estimation with unknown covariance when p is greater than n

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