James-Stein Estimator

Definition

The James–Stein estimator is a biased estimator of the mean of correlated Multivariate Normal Distribution.

Let $\mathbf{z}\sim N_p(\mathbf{\mu },\mathbf{I})$. The James–Stein estimator is defined as ${\hat{\mathbf{\mu }}}_{JS}=\left(1-\frac{p-2}{{\mathbf{z}}^{⊺}\mathbf{z}}\right)\mathbf{z}$

The Least Squares Estimator ${\hat{\mathbf{\mu }}}_{LS}=\mathbf{z}$ is MLE and UMVUE. However, in terms of Mean Squared Error, James-Stein estimator ${\hat{\mathbf{\mu }}}_{JS}$ is better than the least squares estimator ${\hat{\mathbf{\mu }}}_{LS}$ for $p\ge 3$ cases.