Tutorial - Larry Goldstein: The Many Faces of a Simple Identity

Stein's identity states that a mean zero, variance one random variable W has the standard normal distribution if and only if E[Wf(W)] = E[f'(W)] for all f in F, where F is the class of functions such that both sides of the equality exist. Stein (1972) used this identity as the key element in his novel method for proving the Central Limit Theorem. The method generalizes to many distributions beyond the normal, allows one to operate on random variables directly rather than through their transforms, provides non-asymptotic bounds to their limit, and handles a variety of dependence. In addition to distributional approximation, Stein's identity has connections to concentration of measure, Malliavin calculus, and statistical procedures including shrinkage and unbiased risk estimation.