An example of Stein's paradox

Charles Stein showed in 1958, that a nonlinear, biased estimator of a multivariate mean has a lower MSE compared to the ML estimator.

Given a sample of $N$ measurements of $X\sim\mathcal{N}(\mu,\sigma I_p)$ with unknown parameter vector $\mu$ of length $p$.

The James-Stein estimator is given by

$$\begin{equation*} \hat{\mu}_{JS}=\left (1-\frac{(p-2)\frac{\sigma^2}{N}}{\|\bar{x} \|^2}\right ) \bar{x} \end{equation*}$$
where $\bar{x}$ is the sample mean.

This estimator dominates the MLE everywhere in terms of MSE.  For all $\mu\in\mathbb{R}^p$,

$$\begin{equation*} \mathbb{E}_\mu \| \hat{\mu}_{JS}-\mu\|^2 < \mathbb{E}_\mu \| \hat{\mu}_{MLE}-\mu\|^2 \end{equation*}$$
This makes the MLE inadmissible for $p\ge3$!