Exponential Consistency of the M-estimators of Regression Coefficients with Multivariate Responses

In this brief note, we present the exponential consistency of the M-estimators of regression coefficients for models with multivariate responses. We first prove a exponential tail bound for the $\ell_2$-norm of the M-estimator from the true value of the regression coefficients under suitable assumption, which directly leads to the exponential consistency result for the M-estimators. We are working on to apply this general results for some particular M-estimators, including the maximum likelihood estimator, under the special set-ups of multivariate linear regression models and linear mixed-effects models.