Stein's lemma

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Stein's lemma, named in honor of Charles Stein, is 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.

[edit] 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

$E\bigl(g(X)(X-\mu)\bigr)=\sigma^2 E\bigl(g'(X)\bigr).$

In general, suppose X and Y are jointly normally distributed. Then

$\operatorname{Cov}(g(X),Y)=E(g'(X)) \operatorname{Cov}(X,Y).$

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

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

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

${1\over\sigma}\varphi\left({x-\mu \over \sigma}\right).$

Then use integration by parts.

[edit] More general statement

Suppose X is in an exponential family, that is, X has the density

f η (x) = exp(η' T (x) - Ψ(η)) h (x).

Suppose either this density has support $\mathbb{R}$ , or it has support $(a, b)$ and $\exp(\eta'T(x))h(x)\rightarrow 0$ as $x\rightarrow a\text{ or }b$ .