A complete sufficient dimension folding theory with novel methods

这一建议旨在开发一个通用的公式和相关的方法，充分的维度折叠，其中预测是矩阵/数组值，其中一个特定的功能（或参数）的条件分布是感兴趣的。在过去的二十年里，向量值预测器的充分降维方法得到了蓬勃的发展，并积累了惊人的成功应用记录。然而，许多数据是矩阵/数组值，对应用于这些数据的向量值预测器进行足够的降维将失去其充分性和结构，导致解释困难，并且在很大程度上，这些方法将条件分布视为感兴趣的对象，而不区分感兴趣的参数和讨厌的参数。研究者提出了一个矩阵/数组值预测器的足够维度折叠的新范式，该范式专注于条件分布的函数，它可以是涵盖大多数应用的非常广泛的类中的任何一个。此外，研究者建议开发一套连贯的相关技术，用于估计、计算和渐近推断。最近，产生大量复杂高维数据的高通量技术在商业、政府管理、环境研究、机器学习和生物信息学等不同领域越来越普遍。这为统计界发展新的理论和方法提供了相当大的动力，这些理论和方法能够从高维、复杂的结构和大量数据中发现关键证据。充分维度折叠是统计研究的一个新领域，在这些新需求中兴起，并受到这些新需求的推动。研究者建议制定足够维度折叠的理论和方法，以便它们可以专门针对要估计的目标进行定制。这个新范式不仅综合、扩展和深化了充分维折叠的最新进展，而且通过遵循充分性、效率、信息、利益参数和干扰参数的传统，将对充分维折叠的理解与经典统计推断理论相媲美，这些都是帮助推动经典推断走向成熟的关键思想。

This proposal is aimed at developing a general formulation and the related methods for sufficient dimension folding where predictors are matrix-/array- valued, and where a specific functional (or parameter) of the conditional distribution is of interest. The past two decades have seen vigorous development of the sufficient dimension reduction methods for vector-valued predictors, and have accrued a striking record of their successful applications. However, many data are matrix-/array-valued, sufficient dimension reduction for vector-valued predictors applying to such data will lose its sufficiency and structure, resulting difficulties in interpretation, and to a large extent these methods treat the conditional distribution as the object of interest, without discriminating between parameter of interest and nuisance parameter. The investigator proposes a new paradigm for sufficient dimension folding for matrix-/array-valued predictors that focuses on a functional of the conditional distribution, which can be any one in a very wide class that covers most of applications. In addition, the investigator proposes to develop a coherent collection of associated techniques for estimation, computation, and asymptotic inference. Recently, high throughput technologies that produce massive amount of complex and high-dimensional data are increasingly prevalent in such diverse areas as business, government administration, environmental studies, machine learning, and bioinformatics. These provide considerable momentum in the Statistics community to develop new theories and methodologies, that are capable of discovering critical evidence from high-dimensional, complex structural and massive data. Sufficient Dimension Folding is a new area of statistical research that arose amidst, and has been propelled by, these new demands. The investigator proposes to formulate the theories and methodologies of sufficient dimension folding so that they can be specifically tailored to target to be estimated. This new paradigm not only synthesizes, broadens, and deepens the recent advances in sufficient dimension folding, but brings the understanding of sufficient dimension folding on a par with classical statistical inference theory, by following the tradition of sufficiency, efficiency, information, parameter of interests, and nuisance parameters, which are the key ideas that had helped to propel classical inference to its maturity.