Stein's method and the zero bias transformation with application to simple random sampling

Let $W$ be a random variable with mean zero and variance $\sigma^2$. The distribution of a variate $W^*$, satisfying $EWf(W)=\sigma ^2 Ef'(W^*)$ for smooth functions $f$, exists uniquely and defines the zero bias transformation on the distribution of $W$. The zero bias transformation shares many interesting properties with the well known size bias transformation for non-negative variables, but is applied to variables taking on both positive and negative values. The transformation can also be defined on more general random objects. The relation between the transformation and the expression $wf'(w)-\sigma^2 f''(w)$ which appears in the Stein equation characterizing the mean zero, variance $\sigma ^2$ normal $\sigma Z$ can be used to obtain bounds on the difference $E\{h(W/\sigma)-h(Z)\}$ for smooth functions $h$ by constructing the pair $(W,W^*)$ jointly on the same space. When $W$ is a sum of $n$ not necessarily independent variates, under certain conditions which include a vanishing third moment, bounds on this difference of the order $1/n$ for classes of smooth functions $h$ may be obtained. The technique is illustrated by an application to simple random sampling.