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

Stein的方法是一种强大的技术，可以根据概率度量确定两个概率分布之间的距离。它最初是由查尔斯·斯坦在1972年为正态近似开发的。从那时起，斯坦的方法被扩展到许多其他分布极限，包括泊松分布、二项分布和指数分布，并在数学科学中得到了许多应用。最近，用Stein的方法证明了一个数量极限定理的框架，其中极限分布可以表示为多元正态随机向量的光滑函数。也就是说，极限随机向量的形式是g(W)，极限随机向量的形式是g(Z)，其中g是一个光滑函数，W是一个随机向量，它很好地近似于一个多元随机向量Z。许多最具概率性的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。本项目将探索多元正态随机向量函数的斯坦因方法的发展所开辟的富有成效的研究方向。项目的第一阶段将包括对现有理论适用的函数g类进行扩展：学生将把理论扩展到导数为指数增长的函数向量值函数g。学生将提出关于平滑测试函数度量的g(W)和g(Z)分布之间距离的几个一般界限和渐近展开式（控制误差项），在W是随机向量的标准化和的情况下。在g是偶函数的情况下或在一定的匹配矩假设下，预计会出现更快的收敛速度。这项工作将概括斯坦方法文献中的几个重要结果。在数理统计中广泛应用的多元delta方法中，将使用一般界和渐近展开式来得到显式误差界和渐近展开式。这将是第一次详细研究delta方法的收敛速度。特别是，学生将获得“比预期更快”的n^{-1}阶（或更快）收敛速率的充分条件。在项目的后期，学生将推导多元正态随机向量函数在至少一个复杂依赖结构下的分布近似的一般方法。作为本理论的一个重要应用，学生将通过一个重要统计量的极限分布推导出其分布近似的显式误差界。这些结果对统计学家和应用研究人员很有用，他们可以从理论上证明在执行统计测试时使用的各种经验法则，并可能对控制趋同率的因素有新的见解。学生可以考虑的统计量的一个例子是用于无对齐序列比较的d2和d2 ^*统计量。

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.