Stein's lemma

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Stein's lemma, named in honor of Charles Stein, may be characterized as a theorem of probability theory that is of interest primarily because of its application to statistical inference — in particular, its application to James-Stein estimation and empirical Bayes methods.

Statement of the lemma

Suppose X is a normally distributed random variable with expectation μ and variance σ². Further suppose g is a function for which the two expectations E( g(X) (X − μ) ) and E( g ′(X) ) both exist (the existence of the expectation of any random variable is equivalent to the finiteness of the expectation of its absolute value). Then

<math>E(g(X)(X-\mu))=\sigma^2 E(g'(X)).</math>

In order to prove this lemma, recall that the probability density function for the normal distribution with expectation 0 and variance 1 is

<math>\varphi(x)={1 \over \sqrt{2\pi}}e^{-x^2/2}</math>

and that for a normal distribution with expectation μ and variance σ² is

<math>{1\over\sigma}\varphi\left({x-\mu \over \sigma}\right).</math>

Then use integration by parts.