Statistics Colloquium

Envelopes (Cook et al., 2010) were originally proposed for multivariate regression analysis to achieve potential efficiency gains. In this talk, we introduce envelope models for efficient estimation in matrix-variate analysis, where the response Y is a random matrix and the predictor X can be either a scalar, or a vector, or a matrix, treated as non-stochastic. The matrix-variate regression models and their envelope structures are generalizations of conventional regressions and their envelopes. They can be applied to neuroimaging, cross-over design analysis, and multivariate growth curve modeling, among others. Under the envelope framework, redundant information can be eliminated in estimation and the number of parameters can be notably reduced when the matrix-variate dimension is large. Therefore, the estimation can be much more accurate. We further investigate the asymptotic properties of the envelope estimators and show their efficiency in both theoretical and numerical studies.