Stein's method for functions of multivariate normal random vectors: asymptotic expansions, rates of convergence and applications

多元正态随机向量函数的 Stein 方法：渐近展开、收敛速度和应用

• 批准号: 2481303
• 金额: --
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2481303
• 关键词: Stein; method; functions; multivariate; normal

Stein's method is a powerful technique for bounding the distance between two probability distributions with respect to a probability metric. It was originally developed for normal approximation by Charles Stein in 1972. Since then, Stein's method has been extended to many other distributional limits, including the Poisson, binomial and exponential distributions and has found numerous applications throughout the mathematical sciences.Recently, a framework has been developed for using Stein's method to prove quantitative limit theorems in which limiting distribution can be expressed as a smooth function of a multivariate normal random vector. That is the prelimit random vector is of the form g(W) and the limit random vector is of the form g(Z), where g is a smooth function and W is a random vector that is well approximated by a multivariate random vector Z. Many of the most probabilistic limit theorems fall into this class, including the central limit theorem and the chi square approximation of Pearson's chi-square statistic. This project will explore the fruitful research directions opened be the development of Stein's method for functions of multivariate normal random vectors.The first stage of the project will involve making extensions to the classes of functions g to which the existing theory applies: the student will extend the theory to functions vector-valued functions g with derivatives having exponential growth. The student will present several general bounds and asymptotic expansions (with control on the error term) for the distance between the distributions of g(W) and g(Z), with respect to a smooth test function metric, in the case that W is a standardised sum of random vectors. It is expected that faster convergence rates will occur in the case that g is an even function or under certain matching moments assumptions. This work will generalise several significant results from the Stein's method literature. The general bounds and asymptotic expansions will be used to obtain explicit error bounds and asymptotic expansions in the multivariate delta method, which is widely used in mathematical statistics. This will be the first detailed investigation into rates of convergence in the delta method. In particular, the student will obtain sufficient conditions under which `faster than expected' order n^{-1} (or faster) convergence rates occur.Later in the project, the student will derive general for the distributional approximation of functions of multivariate normal random vectors under at least on complicated dependence structure. As an important application of this theory, the student will derive explicit error bounds on the distributional approximation of an important statistic by its limiting distribution. Such results would be useful for statisticians and applied researchers who would gain a theoretical justification of various rules-of-thumb used in the implementation of statistical tests, and potentially new insights into the factors governing convergence rates. An example of a statistic that the student may consider are the D_2 and D_2^* statistics which are used in alignment-free sequence comparison.

Stein方法是一种强大的技术，用于限制两个概率分布之间关于概率度量的距离。它最初是由Charles Stein在1972年开发的。从那时起，Stein的方法已被扩展到许多其他的分布极限，包括泊松分布，二项分布和指数分布，并发现了许多应用在整个mathematical science.Recently，一个框架已经开发出使用Stein的方法来证明定量极限定理，其中极限分布可以表示为一个多元正态随机向量的光滑函数。也就是说，前极限随机向量的形式为g（W），极限随机向量的形式为g（Z），其中g是光滑函数，W是由多变量随机向量Z很好地近似的随机向量。许多概率极限定理都属于这一类，包括中心极限定理和皮尔逊卡方统计量的卡方近似。本专题将探讨多元正态随机向量函数的Stein方法的发展所开启的富有成果的研究方向。本专题的第一阶段将涉及扩展现有理论所适用的函数类g：学生将理论扩展到函数向量值函数g，其导数具有指数增长。学生将提出几个一般的界限和渐近展开（误差项控制）之间的距离g（W）和g（Z）的分布，相对于一个光滑的测试函数度量，在W的情况下，是一个标准化的随机向量的总和。预计在g是偶函数或在某些匹配矩假设下的情况下会出现更快的收敛速度。这项工作将概括几个显着的结果从斯坦的方法文献。在数理统计中广泛使用的多元delta方法中，一般界和渐近展开式将被用来获得显式的误差界和渐近展开式。这将是第一次详细调查的速度收敛的三角洲方法。特别地，学生将获得n^{-1}阶（或更快）收敛速度比预期更快的充分条件。在专题的后面部分，学生将推导多元正态随机向量函数在至少一个复杂相依结构下的分布逼近的一般公式。作为这一理论的一个重要应用，学生将推导出一个重要统计量的分布近似的显式误差界。这样的结果将是有用的统计学家和应用研究人员谁将获得在统计测试的实施中使用的各种经验法则的理论依据，并可能新的见解的因素，收敛速度。学生可以考虑的统计量的一个例子是D_2和D_2^* 统计量，它们用于无干扰序列比较。