Stein's lemma

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

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

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

Then use integration by parts.