Doctoral Dissertation Research: Envelope Models and Methods

多元线性回归(MLR)是研究预测变量和反应变量之间关系的一种范式。它在许多学科中被广泛应用，用于解释反应和预测者之间的相关性或进行对未来结果的预测。随着数据采集和测量技术的发展，许多当代问题涉及到高维数据集。这意味着相当数量的响应信息可能是多余的或不相关的。这种冗余或不相关的数据部分会给MLR中的估计带来变化，使其效率低下。为了解决这个问题，库克等人引入了一类新的模型，称为信封。(2010)。它使用降维技术来识别和提取数据中的相关信息，从而使估计只基于相关部分。然而，到目前为止，信封类仍处于初级阶段。它对数据结构有限制，其优势并不总是能实现的。本博士论文研究项目将研究并使信封类走向成熟，使其更加灵活，并通过用新的模式和方法丰富类来实现进一步的效率收益。将开发解决尺度不变性、异方差和小样本问题的新模型。还将开发新的模式，在现有模式的基础上进一步提高效率。这些对信封类的扩展将对数据结构做出最少的假设，并扩展了信封思想的适用性和威力，使其更具吸引力。这项研究将为社会学、经济学、遗传学和许多其他科学和工程学科提供更有效的数据分析方法。这些方法有望在较小的样本量下达到相同的分析精度，使实验和数据收集过程更短、更容易、成本更低。该项目还将把现有的统计工具，如降维技术和估计大协方差矩阵的方法，以一种新的方式与MLR领域联系起来，开辟其应用的新领域。将开发实施新方法的用户友好的软件。作为博士论文研究改进奖，提供支持使有前途的学生建立一个强大的，独立的研究事业。

Multivariate linear regression (MLR) is a paradigm for studying the relationship between two groups of variables, the predictors and the responses. It is broadly applied in many disciplines for explaining the dependence between the responses and the predictors or for conducting prediction of future outcomes. With the development of technology for data collection and measurement, many contemporary problems involve high-dimensional datasets. This implies the possibility that a considerable amount of the response information may be redundant or irrelevant. This redundant or irrelevant part of the data will bring variation into the estimation in MLR, making it inefficient. To address this problem, a new class of models called envelopes was introduced by Cook et al. (2010). It uses dimension-reduction techniques to identify and extract the relevant information in the data, so that the estimation is based on only the relevant part. Up to this point, however, the envelope class is still in its infancy. It has restrictions on the data structure, and its advantages cannot always be realized. This doctoral dissertation research project will study and bring the envelope class to maturity, making it more flexible and achieving further efficiency gains by enriching the class with new models and methods. New models that address scale invariance, heteroscedasticity, and small sample size issues will be developed. New models that lead to further efficiency gains beyond the current models also will be developed. These extensions of the envelope class will make minimal assumptions on data structure and extend the applicability and power of the enveloping idea, making it more appealing.This research will result in more efficient data analysis methods for sociology, economics, genetics, and many other disciplines in science and engineering. These methods are expected to achieve the same accuracy in analysis with a smaller sample size, making experiments and the data collection process shorter, easier, and less expensive. The project also will link existing statistical tools, such as dimension reduction techniques and methods for estimating large covariance matrix, to the field of MLR in a novel way that opens new frontiers of their application. User-friendly software will be developed that implements the new methodology. As a Doctoral Dissertation Research Improvement award, support is provided to enable a promising student to establish a strong, independent research career.