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

本建议书由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。变点问题出现在各种各样的科学检测时，如果按时间顺序排列的数据序列可以被视为均匀的随机意义。测试数据序列的（理论）均值（平均值）的变化是其他变点问题可以减少的一种基本情况。 建议的第1）a）部分是处理最多一个这样的变化时，数据是随机独立的，并遵循足够一般的条件的非参数检验。这项研究将为从业人员提供关于最多一次平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提出的用于多变量和其他类型数据的此类检验类型。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，这使得这些模型中的统计推断复杂化。提案的第1）B）部分是制定程序，用于在从这些模型中收集数据时，对线性EIVM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发生在EIVM的许多应用的数据分析中，这些应用几乎包括所有的研究领域，但尚未被研究。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。Student t统计量在推理统计学中起着核心作用。研究它何时是渐近标准正态对于各种应用都很重要，例如从大样本数据中构建总体均值的近似置信区间。建议的第2）B）部分的多元Student（随机）过程（多元Student 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，用于某些多元数据序列中均值的可能变化。对于线性EIVM在一些新的条件下，第3）部分的建议是获得渐近结果的推断，从大样本的数据与可能无穷大的方差，从这些模型。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。它们将导致第一次，容易获得的大样本近似置信区域/区间的斜率和截距在这样的EIVM的。

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.