Stein's lemma

Stein's lemma,^[1] 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

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.^[2][3]

See also

• Taylor expansions for the moments of functions of random variables

References

1. Ingersoll, J., Theory of Financial Decision Making, Rowman and Littlefield, 1987: 13-14.
2. Hamada, Mahmoud; Valdez, Emiliano A. (2008). "CAPM and option pricing with elliptically contoured distributions". The Journal of Risk & Insurance. 75 (2): 387–409. doi:10.1111/j.1539-6975.2008.00265.x.
3. Landsman, Zinoviy; Nešlehová, Johanna (2008). "Stein's Lemma for elliptical random vectors". Journal of Multivariate Analysis. 99 (5): 912––927. doi:10.1016/j.jmva.2007.05.006.