Stein's Method - The Basic Approach - The Stein Operator

The Stein Operator

We assume now that the distribution is a fixed distribution; in what follows we shall in particular consider the case where is the standard normal distribution, which serves as a classical example of the application of Stein's method.

First of all, we need an operator which acts on functions from to the real numbers, and which 'characterizes' the distribution in the sense that the following equivalence holds:

 (2.1)quad E (mathcal{A}f)(Y) = 0text{ for all } f quad iff quad Y text{ has distribution } Q.

We call such an operator the Stein operator. For the standard normal distribution, Stein's lemma exactly yields such an operator:

 (2.2)quad Eleft(f'(Y)-Yf(Y)right) = 0text{ for all } fin C_b^1 quad iff quad Y text{ has standard normal distribution.}

thus we can take

 (2.3)quad
(mathcal{A}f)(x) = f'(x) - x f(x)

We note that there are in general infinitely many such operators and it still remains an open question, which one to choose. However, it seems that for many distributions there is a particular good one, like (2.3) for the normal distribution.

There are different ways to find Stein operators. But by far the most important one is via generators. This approach was, as already mentioned, introduced by Barbour and Götze. Assume that is a (homogeneous) continuous time Markov process taking values in . If has the stationary distribution it is easy to see that, if is the generator of, we have for a large set of functions . Thus, generators are natural candidates for Stein operators and this approach will also help us for later computations.

Read more about this topic:  Stein's Method, The Basic Approach

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