Optimal sufficient dimension reduction for the multivariate conditional mean in multivariate regression

The goal of this dissertation is sufficient dimension reduction in multivariate regression. Mainly we want to replace the responses and the predictors in a multivariate regression of Y ∈ Rr |X ∈ Rp by lower-dimensional linearly transformed responses and predictors without loss of information about the multivariate conditional mean E(Y|X).; Focusing on the multivariate conditional mean, Cook and Setodji (2003) recently provided a methodology for sufficient dimension reduction in multivariate regression by estimating the multivariate central mean space. Their test statistics for the dimension of the multivariate central mean subspace is a weighted sum of independent chi-square random variables. The first part of this thesis is devoted to developing an optimal version of their methodology in roughly the same context. The dimension test statistic for the optimal method has an asymptotic chi-square distribution under a hypothesized dimension of the multivariate central mean subspace and the estimate of the multivariate central mean subspace is asymptotically efficient. Additionally, the optimal version allows tests of predictor effects with chi-squared distributions. The comparison between the proposed optimal version and Cook and Setodji (2003) is studied. For this, simulation and a real data set are used to illustrate various methods.; In the second part of the thesis, we develop a methodology for reducing dimensions of the responses. We seek to replace the original response vector by a lower-dimensional linearly transformed response vector without loss of information on the multivariate conditional mean E(Y|X ). For this we define a response mean subspace and the central response mean subspace. We here propose a methodology to do inference about the central response mean subspace. With simple chi-squared tests the methodology enables us to reduce dimensions of predictors and responses simultaneously and to reduce dimension of responses only and provides an asymptotically efficient estimate for the central response mean subspace. Besides, we can perform response effect tests for the multivariate conditional mean. Simulation and the data set used in the first part are studied to illustrate the proposed method.