Envelopes for efficient multivariate parameter estimation

Abstract

Multivariate linear regression of response Y on predictor X is a cornerstone of multivariate statistics. When the dimensions of responses and predictors are not small, it is widely recognized that reducing the dimensionalities of X and Y may often result in improved performance.Cook, Li and Chiaromonte (2010) proposed a new statistical concept{envelopes for increasing efficiency in estimation and prediction in multivariate linear regression. The idea is to envelope the information in the data that is material to the estimation of the parameters of interest, while excluding the information that is immaterial to estimation. This is achieved by estimating an envelope, which is essentially a targeted dimension reduction subspace for particular parameters of interest, to reduce the dimensionality of original problems. In this dissertation, we first propose a fast and stable 1D algorithm for envelope estimation in general. Because envelope estimation involves Grassmann manifold optimizations, our algorithm largely lessens the computational burdens of past and future envelope methods. We then naturally propose two new envelope methods for simultaneously educing X and Y, and for combining envelopes with reduced-rank regression. At the final chapter, we extend the idea of envelope beyond multivariate linear model to rather arbitrary multivariate estimation problems. We propose a constructive definition and a unied framework for incorporating envelopes with many future applications.