Stein's lemma

Luncat ka: pituduh, paluruh

 Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantosanna diantos kanggo narjamahkeun.

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

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

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

${\displaystyle \varphi (x)={1 \over {\sqrt {2\pi }}}e^{-x^{2}/2}}$

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

${\displaystyle {1 \over \sigma }\varphi \left({x-\mu \over \sigma }\right).}$

Then use integration by parts.