Invariance principles and change-point analysis for error-in-variables models

提案标题：变量中误差模型的不变性原理和变点分析变量中误差模型(EIVM)构成了一个有130多年历史的统计学领域。在这些模型中，假设两个或更多感兴趣的变量具有某种形式的关系，并且它们是在测量误差的情况下观察到的。这使得对这种关系的未知参数的推断变得复杂。EIVM也称为测量误差模型，或具有自变量误差的回归模型。在EIVM的许多应用中，对数据分析的需求日益增长，这推动了对EIVM的研究，这些应用几乎涵盖了所有科学和技术领域，例如医学、农业和计量经济学。我的研究项目主要是在大样本观测的基础上，对EIVM的统计推断进行新的渐近研究。同时，这些研究提出并基本上是基于解决概率论领域中关于独立随机变量的自归一化/学生化部分和过程的一些公开问题。因此，这项提案的预期意义在于通过这些研究领域之间的密切相互作用，在统计推断和概率理论方面确立新的方法和结果。独立随机变量的自归一化/学生化部分和过程的不变性原理是EIVM中的一个新概念，它为EIVM中的基于数据的推理提供了一个来源，例如，这些模型中未知参数的易于获得或容易导出的大样本近似置信度区间/区域。它们还允许在一些新的、最一般的条件下研究EIVM。此外，它们在我提出的研究的主要方向之一中发挥着关键作用，即开发研究EIVM中的变点问题的程序。当一个人在收集数据以进行统计推断时，当一个人关注EIVM中可能发生的变化时，这些问题在应用中自然会出现。

Title of Proposal: Invariance Principles and Change-Point Analysis for Error-in-Variables ModelsError-in-variables models (EIVM's) constitute an over 130-year-old area of Statistics. In these models two or more variables of interest are assumed to have a certain form of a relationship, and they are observed with measurement errors. This complicates inferring about unknown parameters of this relationship. EIVM's are also known as measurement error models, or regression models with errors in independent variables. Studies of EIVM's have been stimulated by an increasing demand of data analysis in numerous applications of EIVM's that include virtually every area of science and technology, such as medicine, agriculture and econometrics, for example. My research program primarily deals with new asymptotic studies for statistical inference in EIVM's, based on large samples of observations. At the same time, these studies present, and are substantially based on solving, some open problems in the areas of Probability Theory on self-normalized/Studentized partial sums and partial sums processes of independent random variables. Consequently, the anticipated significance of this proposal amounts to establishing new methodologies and results both in statistical inference and probability theory by means of a close interplay between these areas of research. Invariance principles for self-norma- lized/Studentized partial sums and partial sums processes of independent random variables are new in EIVM's. They provide a source for data-based inference in EIVM's, like, for example, readily available or easily derivable large-sample approximate confidence intervals/regions for unknown parameters in these models. They also allow studying EIVM's under some new, most general conditions. Moreover, they play a crucial role in one of the main directions of my proposed research, namely, in developing procedures for studying change-point problems in EIVM's. Such problems arise naturally in applications when one is concerned with possible changes that may have occurred in an EIVM while collecting data for statistical inference therein.