Collaborative Research: A Paradigm for Dimension Reduction with Respect to a General Functional

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