A complete sufficient dimension folding theory with novel methods

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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.