  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.
