In the words of one of my professors, "Stein's Paradox may very well be the most significant result in Mathematical Statistics since World War II." The problem is this: You observe $X_1, \ldots, X_n \sim \mathcal{N}_p(\mu, \sigma^2 I_p)$, with $\sigma^2$ known, and wish to estimate the mean vector $\mu \in \mathbb{R}^p$. The obvious thing to do, of course, is to use the sample mean $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$ as an estimator of $\mu$. Stein's Paradox is the counterintuitive fact that in dimension $p \ge 3$, this estimator is inadmissible under squared error loss.
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