Collaborative Research: A paradigm for dimension reduction with respect to a general functional

本研究的目的是为条件分布的特定泛函（或参数）提供一个充分降维（SDR）的一般公式和相关方法。过去二十年来，特别提款权方法蓬勃发展，取得了令人瞩目的成功应用记录。然而，这些方法在很大程度上将条件分布作为感兴趣的对象，没有区分感兴趣的参数和讨厌的参数。虽然有针对统计函数的方法，但它们是特定于所考虑的参数的，因此很难应用于其他参数。研究人员提出了一种新的SDR范式，该范式侧重于条件分布的函数，它可以是涵盖大多数应用的非常广泛的类别中的任何一个。此外，研究人员建议开发一套连贯的相关技术，用于估计、计算和渐近推断。产生大量复杂和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，并重新制定现有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和大量数据中发现关键证据。SDR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。研究人员建议重新制定SDR的理论和方法，使它们能够专门针对要估计的目标进行调整。这一新范式不仅综合、拓宽和深化了SDR的最新进展，而且通过遵循充分性、效率性、信息性、利益参数和妨害参数的传统，将对SDR的理解与经典统计推断理论相媲美，这些都是有助于推动经典推断走向成熟的关键思想。

The proposed research aims to developing a general formulation and the related methods for sufficient dimension reduction (SDR) where a specific functional (or parameter) of the conditional distribution is of interest. The past two decades have seen vigorous development of the SDR methods and have accrued a striking record of their successful applications. However, to a large extent these methods treat the conditional distribution as the object of interest, without discriminating between parameter of interest and nuisance parameter. While there are methods that target statistical functionals, they are specific to the parameter in consideration and as such are difficult to apply to other parameters. The investigators propose a new paradigm for SDR 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 investigators propose to develop a coherent collection of associated techniques for estimation, computation, and asymptotic inference.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, and to reformulate the existing ones, that are capable of discovering critical evidence from high-dimensional and massive data. SDR is a recent area of statistical research that arose amidst, and has been propelled by, these new demands. The investigators propose to reformulate the theories and methodologies of SDR 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 SDR, but brings the understanding of SDR 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 has helped to propel classical inference to its maturity.