Stein's lemma

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 application to statistical inference — in particular, its application to James-Stein estimation and empirical Bayes methods.

tatement 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

:Eigl(g(X)(X-mu)igr)=sigma^2 Eigl(g'(X)igr).

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{2pie^{-x^2/2}

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

:{1oversigma}varphileft({x-mu over sigma} ight).

Then use integration by parts.


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