Proof of Stein's example

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Stein's example is an important result in decision theory which can be stated as

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.

Sketched proof[edit]

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,

(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

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.