Estimation of multivariate means with heteroscedastic errors using envelope models

In this article, we propose envelope models that accommodate heteroscedastic error structure in the framework of estimating multivariate means for different populations. Envelope models were introduced by Cook, Li, and Chiaromente (2010) as a parsimonious version of multivariate linear regression that achieves efficient estimation of the coefficients by linking the mean function and the covariance structure. In the original development, constant covariance structure was assumed. The heteroscedastic envelope models we propose are more flexible in allowing a more general covariance structure. Their asymptotic variances and Fisher consistency are studied. Simulations and data examples show that they are more efficient than standard methods of estimating the multivariate means, and also more efficient than the envelope model assuming constant covariance structure.