In this paper we derive the optimal linear shrinkage estimator for the large-dimensional mean vector using random matrix theory. The results are obtained under the assumption that both the dimension $p$ and the sample size $n$ tend to infinity such that $n^{-1}p^{1-\gamma} \to c\in(0,+\infty)$ and $\gamma\in [0, 1)$. Under weak conditions imposed on the the underlying data generating process, we find the asymptotic equivalents to the optimal shrinkage intensities, prove their asymptotic normality, and estimate them consistently. The obtained non-parametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically with probability $1$ the quadratic loss in the case of $c\in(0,1)$. For $c\in(1,+\infty)$ we modify the suggested estimator by using a feasible estimator for the precision covariance matrix. At the end, an exhaustive simulation study and an application to real data are provided where the proposed estimator is compared with known benchmarks from the literature.
↧