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A James-Stein correction

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You observe $5$ independent means $x_{i}$ , each normal with variance $1$ around its own true value, and $\sum x_{i}^{2} = 30$ . The James-Stein estimator toward $0$ multiplies every $x_{i}$ by $1 - (p - 2) / \sum x_{i}^{2}$ , where $p$ is the number of means. What does it report for the mean observed at $4$ ? Give a fraction.

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Hint

Compute the common factor first, then apply it to $4$ .

Solution

With $p = 5$ , the numerator is $5 - 2 = 3$ .
The factor is $1 - \frac{3}{30} = \frac{9}{10}$ .
The estimate is $\frac{9}{10} \times 4 = \frac{18}{5}$ .

Answer: $\frac{18}{5}$

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