Mathematical Sciences: Stein's Method and the Zero Bias Transformation

摘要在过去的二十年中，Stein的方法已经成为一种越来越有价值的工具，用于用正态或泊松方法求得某些相关随机变量和的分布近似的界。研究人员提出探索一种新的分布变换，创造了“零偏差变换”，以及一种相关的新耦合，可以与Stein的方法结合使用，以获得具有特定依赖结构的随机变量和的正态边界。新的耦合可以用来阐明某些问题，而以前的方法只能产生不完整的图像。此外，当近似对称相关随机变量的总和时，或者那些具有消失的第三矩的随机变量，零偏差耦合是自然的，并且承诺产生更小的近似误差界限。零偏差技术在局部和全局依赖的情况下有许多潜在的应用，例如，非线性抽样和秩统计。此外，零偏差变换可以定义为一般随机对象，如随机测量和扩散，并应用于相关观测的经验中心极限定理，以及沃尔福威茨定理的过程处理。正态曲线广泛用于各种统计应用中，例如为质量控制对大量商品进行抽样时，或在民意调查中。在实际情况中，正态曲线通常只是难以计算的东西的理想化。在抽样的情况下，更大的样本量会提高正态近似的准确性，但抽样通常是昂贵的或耗时的。因此，了解正态曲线如何很好地逼近小样本和现实样本量的真实情况是有用的。正态曲线近似也出现在各种其他，通常更复杂的统计环境中。有许多方法可以评估正态近似的准确性，每一种方法都最适合于特定的需要。除了这些技术之外，研究人员正在开发一种新的方法，可以在某些情况下改进现有的方法。例如，在抽样方面使用新方法将从实际的样本中得出更可靠的统计结论。

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.