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

变量误差模型（Error-in-Variables models， EIVM’s）是一个有着130多年历史的统计学领域。在这些模型中，假设两个或多个感兴趣的变量具有某种形式的关系，并以测量误差观察它们。这使得对这种关系的未知参数的推断变得复杂。EIVM也被称为测量误差模型，或者在自变量中有误差的回归模型。EIVM的大量应用对数据分析的需求日益增长，刺激了EIVM的研究，这些应用几乎涵盖了科学和技术的每个领域，例如医学、农业和计量经济学。我的研究项目主要涉及基于大样本观察的EIVM统计推断的新渐近研究。同时，这些研究提出并实质上是基于解决概率论领域中关于独立随机变量的自规格化部分和和部分和过程的一些开放问题。因此，这项建议的预期意义在于通过这些研究领域之间的密切相互作用，在统计推断和概率论方面建立新的方法和结果。独立随机变量的自归一化/学生化部分和过程和部分和过程的不变性原理是EIVM中的新内容。它们为EIVM中基于数据的推断提供了一个来源，例如，这些模型中未知参数的易于获得或易于推导的大样本近似置信区间/区域。他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起着至关重要的作用，即开发用于研究EIVM中的变化点问题的程序。在应用程序中，当人们在为其中的统计推断收集数据时，关注EIVM中可能发生的变化时，自然会出现此类问题。

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.