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 \(p\ge3\)!