Indirect multivariate response linear regression

We propose a class of estimators of the multivariate response linear regression coefficient matrix that exploits the assumption that the response and predictors have a joint multivariate normal distribution. This allows us to indirectly estimate the regression coefficient matrix through shrinkage estimation of the parameters of the inverse regression, or the conditional distribution of the predictors given the responses. We establish a convergence rate bound for estimators in our class and we study two examples, which respectively assume that the inverse regression's coefficient matrix is sparse and rank deficient. These estimators do not require that the forward regression coefficient matrix is sparse or has small Frobenius norm. Using simulation studies, we show that our estimators outperform competitors.