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.

翻译：在本简短说明中,我们展示了具有多变反应模型回归系数M-估计指数的指数一致性。我们首先证明M-估计指数从适当假设的回归系数真实值中得出的以美元=2美元-norm为单位的指数尾巴,这直接导致M-估计指数的指数一致性结果。我们正努力在多变线性回归模型和线性混合效应模型的特殊组合下,将这一总体结果应用于某些特定的M-估计指数,包括最大可能性估测器。