ENVELOPE MODEL FOR MULTIVARIATE LINEAR REGRESSION WITH ELLIPTICAL ERROR

Abstract

In recent years, the need for models which can accommodate higher order covariates have increased greatly. We first consider linear regression with vector-valued response Y and tensor-valued predictors X. Envelope models (Cook et al., 2010) can significantly improve the estimation efficiency of the regression coefficients by linking the regression mean with the covariance of the regression error. Most existing tensor regression models assume that the conditional distribution of Y given X follows a normal distribution, which may be violated in practice. In Chapter 2, we propose an envelope multivariate linear regression model with tensor-valued predictors and elliptically contoured error distributions. The proposed estimator is more robust to violations of the error normality assumption, and it is more efficient than the estimators without considering the underlying envelope structure. We compare the new proposal with existing estimators in extensive simulation studies. In Chapter 3, we explore how the missing data problem can be addressed for multivariate linear regression setting with envelopes and elliptical error. A popular and efficient approach, multiple imputation is implemented with bootstrapped expectation-maximization (EM) algorithm to fill the missing data, which is then followed with an adjustment in estimating regression coefficients. Simulations with synthetic data as well as real data are presented to establish the superiority of the adjusted multiple imputation method proposed.