Stein's lemma

Citakan:Tarjamahkeun Stein's lemma, ngaran keur ngahargaan ka Charles Stein, may be characterized as a théorem 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 nilai ekspektasi μ and varian 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 finiténess of the expectation of its absolute value). Then

$E(g(X)(X-\mu ))={\sigma }^{2}E(g^{\prime }(X)).$

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

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

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

$\frac{1}{\sigma }\phi \left(\frac{x-\mu }{\sigma }\right).$

Then use integration by parts.