Proof of Stein's example
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- The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk in dimension at least 3.
The following is an outline of its proof. The reader is referred to the main article for more information.
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:
(This result is known as Stein's lemma.)
Now, we choose
If met the "well-behaved" condition (it doesn't, but this can be remedied -- see below), we would have
Then returning to the risk function of :
This quadratic in is minimized at
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.