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

9505075 Goldstein摘要 在过去的二十年里，Stein的方法已经成为一个越来越有价值的工具，用于获得正常或泊松分布近似的某些相依随机变量的总和的界限。研究人员建议探索一种新的分布变换，称为“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，用于获得具有特定依赖结构的随机变量和的正常范围。新的耦合可以用来阐明某些问题，以前的方法只产生了一个不完整的图片。此外，当近似的对称相依随机变量的总和，或那些具有消失的三阶矩，零偏耦合是自然的，并承诺产生更小的边界上的近似误差。零偏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，零偏变换可以定义为一般随机对象，如随机测度和扩散，应用于经验中心极限定理的相关观测，和沃尔福威茨定理的过程处理。 正态曲线广泛应用于各种统计应用中，例如为质量控制或民意调查而对大量商品进行抽样时。在真实的情况下，法向曲线往往只是难以计算的东西的理想化。在采样的上下文中，更好的正态近似精度伴随着更大的样本量，但采样通常是昂贵或耗时的。因此，了解正态曲线在多大程度上接近小样本和实际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的，往往更复杂的统计背景。存在许多用于评估正态近似的准确性的方法，每种方法最适合于特定的需要。除了这些技术，研究人员正在开发一种新的方法，在某些情况下改进现有的方法。例如，在抽样中使用新方法，将从实际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.