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

这项建议由1)-3)部分组成，分别涉及标题中列出的统计学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在这些领域取得的最新进展。*在检测按时间顺序排列的数据序列是否在随机意义上被视为同质时，在许多科学中都会出现转折点问题。测试数据序列的(理论)平均值(平均值)的变化是其他变化点问题可以减少到的一种基本情况。建议的1)a)部分是当数据是随机独立的并且遵循足够一般的条件时，处理至多一个这样的变化的非参数检验。这项研究将为实践者提供关于最多一次均值变化的可用检验的性能和使用的指导，并将扩展申请人最近为多变量和其他类型的数据提出的这类检验的类型。在线性变量误差模型(EIVM)中，两个变量是线性相关的，并且观察到测量误差，从而使这些模型中的统计推断复杂化。建议的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中的斜率和截距的可能变化进行非参数检测。所提出的变点问题自然地出现在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，但尚未被研究。*建议的第二部分旨在获得关于多元学生t-统计量的渐近正态性质的特征的猜想的全部或至少相当一般的部分解决方案。学生t统计量在推断统计中起着核心作用。研究它何时是渐近标准正态对于各种应用是很重要的，例如从大样本数据中构造总体平均值的近似置信度区间。*b)建议的多变量学生(随机)过程(多变量学生t统计量的推广)的预期不变性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，并可成为开发对某些多变量数据序列的均值可能变化的非参数检验的关键工具。*对于某些新条件下的线性EIVM，建议的第三部分是从这些模型中可能具有无限方差的大样本数据中获得推断的渐近结果。结果将基于线性回归的最小二乘估计器和独立随机变量的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容易获得的大样本近似置信域/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.