9505075 Goldstein Abstract In the past two decades, Stein's method has become an increasingly valuable tool for obtaining bounds for distributional approximations of certain sums of dependent random variables by the normal or Poisson. The investigators propose to explore a new distributional transformation, coined the ``zero bias transformation,'' and an associated new coupling that may be used in conjunction with Stein's method for obtaining bounds to the normal for sums of random variables having particular dependence structures. The new coupling can be used to shed light on certain questions where previous methods have yielded only an incomplete picture. In addition, when approximating the sum of symmetric dependent random variables, or those having vanishing third moment, the zero bias coupling is natural and promises to yield smaller bounds on the approximation error. The zero bias technique has numerous potential applications to cases of both local and global dependence, for instance, to nonlinear sampling and rank statistics. Moreover, the zero bias transformation may be defined for general random objects such as random measures and diffusions, with applications to Empirical Central Limit Theorems for dependent observations, and a process treatment of the Wald Wolfowitz theorem. The normal curve is used extensively in a variety of statistical applications, such as when sampling a large lot of goods for quality control, or in opinion polling. In real situations the normal curve is often only an idealization of something too difficult to compute. In the context of sampling, better accuracy of the normal approximation comes with larger sample sizes, but sampling is typically expensive or time consuming. Therefore, it is useful to have an idea of how well the normal curve approximates the true situation for small and realistic sample sizes. The normal curve approximation also appears in a variety of other, often more complex, statistical contexts. There exist a number of approaches for assessing the accuracy of the normal approximation, each best suited to a particular need. In adding to these techniques, the investigators are developing a new method which improves on those existing, for some situations. For instance, use of the new method, applied in the sampling context, would result in more reliable statistical conclusions drawn from realistic samples.
