ON TWO NEW ESTIMATORS FOR THE CMS THROUGH EXTENSIONS OF OLS

Abstract

As a useful tool for multivariate analysis, sufficient dimension reduction (SDR) aims to reduce the predictor dimensionality while simultaneously keeping the full regression information, or some specific aspects of the regression information, between the response and the predictor. When the goal is to retain the information about the regression mean, the target of the inference is known as the central mean space (CMS). Ordinary least squares (OLS) is a popular estimator of the CMS, but it has the limitation that it can recover at most one direction in the CMS. In this dissertation, we introduce two new estimators of the CMS: the sliced OLS and the hybrid OLS. Both estimators can estimate multiple directions in the CMS. The dissertation is organized as follows. Chapter 1 provides a literature review about basic concepts and some traditional methods in SDR. Motivated from the popular SDR method called sliced inverse regression, sliced OLS is proposed as the first extension of OLS in Chapter 2. The asymptotic properties of sliced OLS, order determination, as well as testing predictor contribution through sliced OLS are studied in Chapter 2 as well. It is well-known that slicing methods such as sliced inverse regression may lead to different results with different number of slices. Chapter 3 proposes hybrid OLS as the second extension. Hybrid OLS shares the benefit of sliced OLS and recovers multiple directions in the CMS. At the same time, hybrid OLS improves over sliced OLS by avoiding slicing. Extensive numerical results are provided to demonstrate the desirable performances of the proposed estimators. We conclude the dissertation with some discussions about the future work in Chapter 4.