  We consider high-dimensional multivariate linear regression models, where the
joint distribution of covariates and response variables is a multivariate
normal distribution with a bandable covariance matrix. The main goal of this
paper is to estimate the regression coefficient matrix, which is a function of
the bandable covariance matrix. Although the tapering estimator of covariance
has the minimax optimal convergence rate for the class of bandable covariances,
we show that it has a sub-optimal convergence rate for the regression
coefficient; that is, a minimax estimator for the class of bandable covariances
may not be a minimax estimator for its functionals. We propose the blockwise
tapering estimator of the regression coefficient, which has the minimax optimal
convergence rate for the regression coefficient under the bandable covariance
assumption. We also propose a Bayesian procedure called the blockwise tapering
post-processed posterior of the regression coefficient and show that the
proposed Bayesian procedure has the minimax optimal convergence rate for the
regression coefficient under the bandable covariance assumption. We show that
the proposed methods outperform the existing methods via numerical studies.
