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，用于将两个概率分布之间的距离相对于概率度量进行限定。它最初是由查尔斯·斯坦在1972年为正态近似而开发的。从那时起，Stein的方法已经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，并在整个数学科学中得到了大量的应用。最近，人们发展了一个框架来使用Stein方法证明定量极限定理，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，预极限随机向量是g(W)的形式，极限随机向量是g(Z)的形式，其中g是光滑函数，W是被多元随机向量Z很好地逼近的随机向量。许多最概率的极限定理都属于这类，包括中心极限定理和皮尔逊卡方统计量的卡方逼近。本课题将探索多元正态随机向量函数的Stein方法的发展所开启的卓有成效的研究方向。本课题的第一阶段将涉及对现有理论所适用的函数类g的推广：学生将把该理论推广到导数具有指数增长的向量值函数g。在W是随机向量的标准化和的情况下，学生将给出g(W)和g(Z)的分布之间的距离相对于光滑测试函数度量的几个一般界和渐近展开式(控制误差项)。当g为偶数函数或在一定的匹配矩假设下，收敛速度会更快。这项工作将概括斯坦方法文献中的几个重要结果。在数理统计中广泛使用的多元Delta方法中，一般的界和渐近展开式将被用来得到显式的误差界和渐近展开式。这将是第一次对Delta方法的收敛速度进行详细调查。特别地，学生将得到比预期更快的n^{-1}阶(或更快)收敛速度的充分条件。在本项目中，学生将得到至少在复杂相依结构下多元正态随机向量函数的分布逼近的一般公式。作为这一理论的一个重要应用，学生将通过一个重要统计量的极限分布得出其分布近似的显式误差界。这样的结果将对统计学家和应用研究人员有用，他们将从理论上证明在实施统计检验中使用的各种经验法则，并有可能对影响收敛速度的因素有新的见解。学生可以考虑的统计量的一个例子是用于无比对序列比较的D_2和D_2^*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.