Proof of Stein's example


Stein's example is an important result in decision theory which can be stated as
The following is an outline of its proof. The reader is referred to the main article for more information.

Sketched proof

The risk function of the decision rule is
Now consider the decision rule
where. We will show that is a better decision rule than. The risk function is
— a quadratic in. We may simplify the middle term by considering a general "well-behaved" function and using integration by parts. For, for any continuously differentiable growing sufficiently slowly for large we have:
Therefore,
Now, we choose
If met the "well-behaved" condition, we would have
and so
Then returning to the risk function of :
This quadratic in is minimized at
giving
which of course satisfies
making an inadmissible decision rule.
It remains to justify the use of
This function is not continuously differentiable, since it is singular at. However, the function
is continuously differentiable, and after following the algebra through and letting, one obtains the same result.