Nonparametric change-point analysis, invariance principles for multivariate Student processes, and asymptotic theory in linear errors-in-variables models

本提案由 1)-3) 部分组成，分别涉及标题中列出的统计和概率论的三个相互关联的领域。它基于申请人的研究出版物中在这些领域取得的最新进展。 *****当检测按时间顺序排列的数据序列是否可以被视为随机意义上的同质时，各种科学中都会出现变点问题。测试数据序列的（理论）均值（平均值）的变化是一种可以减少其他变点问题的基本情况。 提案的第 1) a) 部分是在数据随机独立并遵循足够普遍的条件时处理最多一个此类变化的非参数检验。这项研究将为从业者提供有关最多均值一次变化的可用测试的性能和使用的指导，并将扩展申请人最近针对多变量和其他类型数据提出的此类测试的类型。******在线性变量误差模型（EIVM）中，两个变量是线性相关的，并且观察到的测量误差使这些模型中的统计推断变得复杂。提案的第 1) b) 部分是开发程序，用于在从这些模型收集数据的同时，对线性 EIVM 的斜率和截距可能发生的变化进行非参数检测。所提出的变点问题在 EIVM 的众多应用的数据分析中自然出现，这些应用几乎包括所有研究领域，但尚未进行研究。********提案的第 2) a) 部分旨在获得关于多元学生 t 统计量的渐近正态性特性的猜想的完整或至少是相当一般的部分解决方案。学生 t 统计量在推论统计中发挥了核心作用。研究何时渐近标准正态对于各种应用都很重要，例如根据大数据样本构建总体均值的近似置信区间。********提案第 2) b) 部分的多元学生（随机）过程（多元学生 t 统计量的推广）的预期不变原理将有助于概率论自归一化过程的极限定理的最新进展，并且可以成为开发非参数检验的关键工具，以检验可能发生的变化一些多元数据序列中的均值。*****对于一些新条件下的线性 EIVM，提案的第 3) 部分是从这些模型可能存在无限方差的大数据样本中获得用于推断的渐近结果。结果将基于线性回归的最小二乘估计量和独立随机变量自归一化过程的不变性原理。它们将首次为此类 EIVM 中的斜率和截距提供易于获得的大样本近似置信区域/区间。

This Proposal consists of parts 1)-3) that deal respectively with three interrelated areas of statistics and probability theory that are listed in the title. It is based on recent advances made in these areas in the applicant's research publications.******Change-point problems arise in a large variety of sciences when detecting if a sequence of chronologically ordered data could be viewed as homogeneous in a stochastic sense. Testing for changes in the (theoretical) mean (average) of a data sequence is one basic situation to which other change-point problems can be reduced. Part 1) a) of Proposal is to deal with nonparametric tests for at most one such change when the data are stochastically independent and follow general enough conditions. This research will provide guidance to practitioners on the performance and use of the available tests for at most one change in the mean and will extend the type of such tests recently proposed by the applicant for multivariate and other kinds of data.******In linear errors-in-variables models (EIVM) two variables are linearly related and are observed with measurement errors that complicate statistical inference in these models. Part 1) b) of Proposal is to develop procedures for nonparametric detection of a possible change in the slope and intercept in linear EIVM's while collecting data from these models. The proposed change-point problems occur naturally in data analysis of numerous applications of EIVM's, which include virtually all research areas, but have not yet been studied.******Part 2) a) of Proposal intends to obtain the full, or at least a rather general partial, resolution of a conjecture on a characterization of the asymptotic normality property of the multivariate Student t-statistic. The Student t-statistic has played a central role in inferential statistics. Studying when it is asymptotically standard normal is important for various applications such as constructing approximate confidence intervals for a population mean from large samples of data.******The anticipated invariance principles for the multivariate Student (stochastic) process (a generalization of the multivariate Student t-statistic) of part 2) b) of Proposal will contribute to recent advances in limit theorems for self-normalized processes of probability theory and can become a key tool for developing nonparametric tests for a possible change in the mean in a sequence of some multivariate data.******For linear EIVM's under some new conditions, part 3) of Proposal is to obtain asymptotic results for inference from large samples of data with possibly infinite variances from these models. The results will be based on the least squares estimators from linear regression and invariance principles for self-normalized processes of independent random variables. They will lead to first time, easily available large-sample approximate confidence regions/intervals for the slope and intercept in such EIVM's.