the james-stein estimator

rob mahurin — Tue, 06 Jan 2026 23:28:58 GMT

I am reminded today by a video by "Mathemaniac" about the James-Stein estimator.

Suppose I'm trying to estimate a number $n$ of independent parameters simultaneously, by taking a sample from each one-dimensional normal distribution with unknown means $\mu_n$ and unit standard deviations $\sigma_n=1$. The naïve estimator is to use each sample $x_n$ as an estimate $\hat\mu_n$ of the mean. However, if my number of parameters is large enough, the zero-biased estimator

$$ \left(\begin{array}{c} \hat \mu_1 \\ \vdots \\ \hat \mu_n \end{array}\right) = \left( 1-\frac{n-2}{x_1^2 + \cdots + x_n^2} \right) \left(\begin{array}{c} x_1 \\ \vdots \\ x_n \end{array}\right) $$

actually produces a smaller mean-squared error on the ensemble as a whole.

mask of the robtasm (Posts about paradox)

the james-stein estimator