Optimal Shrinkage Estimator for High-Dimensional Mean Vector

TL;DR

This paper develops an optimal linear shrinkage estimator for high-dimensional mean vectors using random matrix theory, providing a simple, asymptotically optimal solution that adapts to the ratio of dimension to sample size.

Contribution

It introduces a nonparametric, asymptotically optimal shrinkage estimator for high-dimensional means applicable when both dimension and sample size grow large.

Findings

Estimator minimizes quadratic loss for c in (0,1)

Modified estimator handles c > 1 using precision matrix estimation

Simulation and real data show competitive performance

Abstract

In this paper we derive the optimal linear shrinkage estimator for the high-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 in such a way that $p/n \to c\in(0,\infty)$. Under weak conditions imposed on the underlying data generating mechanism, we find the asymptotic equivalents to the optimal shrinkage intensities and estimate them consistently. The proposed nonparametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically, with probability $1$, the quadratic loss when $c\in(0,1)$. When $c\in(1, \infty)$ we modify the estimator by using a feasible estimator for the precision covariance matrix. To this 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. It turns out that the existing estimators of the mean vector, including the new proposal, converge to the sample mean vector when the true mean vector has an unbounded Euclidean norm.