The principal investigator (P.I.) will study six topics in dimension reduction problems, develop new methodologies and analyze their properties and performances. Topic One concerns central space estimation. The P.I. will initiate a totally different approach from the current literature, using a semiparametric treatment. The new view point results in a complete class of estimators for the central space which contains all possible estimators. In addition, it relaxes various conditions currently required in the existing methods. Finally, it illustrates the relations between various existing estimators and reveals the underlying reason which enables all these estimators to function. Topic Two concerns central mean space estimation. The P.I. will establish parallel results to Topic One, using the same general idea but via different analytic derivation. Topic Three concerns two common conditions in dimension reduction: the linearity condition and the constant variance condition. The P.I. will reveal an astonishing discovery that these conditions are not only redundant, but also detrimental. They are redundant in that consistent estimation can still be carried out of these conditions are relaxed. They are detrimental in two ways. 1. If these conditions are not met, but are falsely assumed, the classical estimators are biased. 2. If these conditions are met and are used, the classic estimators will have inflated variances compared to the same estimator without using these conditions. This research will prompt re-evaluation of these popular conditions and divert the current research trend of further exploiting these conditions. Topic Four concerns statistical inference and efficient estimation of the central space. The P.I. will devise a convenient parameterization and proposeto convert the space estimation problem to an equivalent parameter estimation problem which facilitates statistical inference. The P.I. will also establish the semiparametric efficiency bound for the central space estimation and will derive the optimal efficient estimator. She will further provide theoretical proof of the root-$n$ convergence rate and the efficiency. Topic Five concerns inference and efficient estimation for the central mean space. The P.I. will establish parallel results to Topic Four, illustrate the difference between the two problems and highlight their drastically different analytic results. Topic Six concerns deciding the dimension of the reduced space. The P.I. will propose three methods tailored to different estimation procedures and examine their usefulness.
The series of projects in this proposal will shed new light on the dimension reduction problems and resolve some of the most fundamental and outstanding issues in this field. Accompanying the modern development of sciences and technologies, richer and more complex data have become a common phenomenon. Data sets from medical science, genetics, environmental science, social behavioral science including internet behavior studies easily contain a large amount of variables that dimension reduction is unavoidable. The new methodologies will generate wide interest and have important application in these fields. They will also provoke further studies and development in related semiparametric problems and computing methods in statistical sciences itself.
