Stein's lemma

Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein estimation and empirical Bayes methods — and its applications to portfolio choice theory. The theorem gives a formula for the covariance of one random variable with the value of a function of another, when the two random variables are jointly normally distributed.

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

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

Proof

Main article: Proof of Stein's example § Sketched proof

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.

More general statement

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

$f_\eta(x)=\exp(\eta'T(x) - \Psi(\eta))h(x).$

Suppose this density has support $(a,b)$ where $a,b$ could be $-\infty ,\infty$ and as $x\rightarrow a\text{ or }b$,$\exp (\eta'T(x))h(x) g(x) \rightarrow 0$ where $g$ is any differentiable function such that $E|g'(X)|<\infty$ or $\exp (\eta'T(x))h(x) \rightarrow 0$ if $a,b$ finite. Then

$E((h'(X)/h(X) + \sum \eta_i T_i'(X))g(X)) = -Eg'(X).$

The derivation is same as the special case, namely, integration by parts.

If we only know $X$ has support $\mathbb{R}$, then it could be the case that $E|g(X)| <\infty \text{ and } E|g'(X)| <\infty$ but $\lim_{x\rightarrow \infty} f_\eta(x) g(x) \not= 0$. To see this, simply put $g(x)=1$ and $f_\eta(x)$ with infinitely spikes towards infinity but still integrable. One such example could be adapted from $f(x) = \begin{cases} 1 & x \in [n, n + 2^{-n}) \\ 0 & \text{otherwise} \end{cases}$ so that $f$ is smooth.

Extensions to elliptically-contoured distributions also exist.

See also

• Taylor expansions for the moments of functions of random variables