Several Problems in Dimension Reduction

Dimension reduction is almost an unavoidable practice in the modern era with big and complex data, and is the main topic in this proposal. Benefit from our success in this area, our research projects will further broaden the scope of the dimension reduction research domain and resolve several difficult, subtle and under-studied issues in this field. The results will generate wide interest and prompt applications in different fields where measurements may contain errors and/or where the data may not be completely random such as in medical studies.

The principal investigator (P.I.) will study five topics in dimension reduction problems, develop new methodologies and analyze their properties and performances. The first topic concerns two common conditions in dimension reduction: the linearity condition and the constant variance condition. The P.I. will reveal the true effect of these conditions, understand why the current implementation of these conditions are detrimental, prescribe a procedure to optimally utilize these conditions and show the resulting efficiency gain. This research will change the current practice of implementing these conditions. The second topic concerns central variance space estimation. The P.I. will treat the central variance space simultaneously with the central mean space and demonstrate why it is necessary to do so. She will consider both when the two spaces differ and when they coincide, and establish methods for estimation and inference in both cases. The third topic concerns multivariate responses. The P.I. will extend the central space concept to the general framework using a new way of modeling that serves as a middle ground between two current approaches. She will demonstrate the advantages of the new model and propose estimation and inference methods. The P.I. will also establish the semiparametric efficiency bound for the multivariate response central space estimation,and illustrate why and how to conduct estimation to achieve local efficiency. The fourth topic concerns dimension reduction when covariates contain measurement errors. The P.I. will study a general semiparametric dimension reduction model when the covariate of interest is measured with error and modeled parametrically. A bias-correction procedure will be devised which does not require modeling the unobservable variable distribution. The estimators will be shown to be consistent, asymptotically normal. The fifth topic concerns dimension reduction arising when analyzing case-control data in secondary analysis. The P.I. will illustrate that when multiple covariates are available, despite of a completely parametric modeling of the regression mean function, the case-control nature of the data requires various nonparametric estimation with multivariate covariates, hence leading to the need of reducing dimension. The inherent relation imposed by the original model further leads to dimension reduction with special structure, for which the P.I. will devise consistent estimators and establish their local efficiency and robustness against the misspecification of the regression error distribution.