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。它是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在随机意义上视为同质时，在许多科学中都会出现变化点问题。测试数据序列的（理论）平均值（平均值）的变化是可以减少其他变化点问题的一种基本情况。提案的第1)a)部分是在数据随机独立并遵循一般足够条件的情况下，处理最多一个这样的变化的非参数检验。这项研究将为从业人员提供关于至多针对一种平均值变化的现有测试的性能和使用的指导，并将扩展申请人最近提出的用于多变量和其他类型数据的此类测试的类型。******在线性变量误差模型（EIVM）中，两个变量是线性相关的，并且观测到的测量误差使这些模型中的统计推断复杂化。提案的第1)b)部分是在从这些模型收集数据时，开发线性EIVM中斜率和截距可能变化的非参数检测程序。所提出的变化点问题自然出现在EIVM的许多应用的数据分析中，这些应用几乎涵盖了所有的研究领域，但尚未得到研究。******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的表征的猜想的全部或至少是相当一般的部分解决。学生t统计量在推理统计中起着核心作用。研究何时为渐近标准正态对于各种应用都很重要，例如从大量数据样本中构造总体均值的近似置信区间。******提案第2)b)部分中多元学生（随机）过程（多元学生t统计量的推广）的预期不变性原则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，用于在一些多元数据序列中平均值的可能变化。******对于一些新条件下的线性EIVM，提案的第3部分是从这些模型中获得可能具有无限方差的大样本数据的渐近推断结果。结果将基于线性回归的最小二乘估计和独立随机变量自归一化过程的不变性原理。它们将导致此类EIVM中斜率和截距的第一次，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.