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

9505075戈尔茨坦摘要在过去的二十年里，斯坦的方法已经成为一种越来越有价值的工具，用于获得按正态或泊松分布逼近某些相依随机变量和的界。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，它可以与斯坦的方法结合使用，以获得具有特定相依结构的随机变量和的正态界限。这种新的耦合可以用来阐明某些问题，在这些问题上，以前的方法只能产生一个不完整的画面。此外，当逼近对称相依随机变量之和或具有零三阶矩的随机变量时，零偏耦合是自然的，并且有望产生较小的逼近误差界。零偏技术在局部和全局相关的情况下都有许多潜在的应用，例如，在非线性抽样和秩统计量方面。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，应用于相依观测的经验中心极限定理，以及Wald Wolfowitz定理的过程处理。正态曲线在各种统计应用中被广泛使用，例如在对大量商品进行抽样以进行质量控制时，或在民意调查中。在实际情况中，正常曲线通常只是一些太难计算的东西的理想化。在采样环境中，样本大小越大，正态近似的精度越高，但采样通常比较昂贵或耗时。因此，对于小而真实的样本量，了解法线曲线与真实情况的接近程度是很有用的。正常曲线近似也出现在各种其他的、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精度，每种方法都最适合特定的需要。在增加这些技术的同时，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，使用适用于抽样的新方法，将产生从实际样本中得出的更可靠的统计结论。

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.