Scale invariant and efficient estimation for groupwise scaled envelope model

Motivated by different groups containing different group information under the heteroscedastic error structure, we propose the groupwise scaled envelope model that is invariable to scale changes and is permissible for distinct regression coefficients and the heteroscedastic error structure across groups. It retains the potential of the scaled envelope methods to keep the scale invariant and allows for both different regression coefficients and different error structures for diverse groups. Further, we demonstrate the maximum likelihood estimators and its theoretical properties including parameter identifiability, asymptotic distribution and consistency of the groupwise scaled envelope estimator. Lastly, simulation studies and a real-data example demonstrate the advantages of the groupwise scaled envelope estimators, including a comparison with the standard model estimators, groupwise envelope estimators, scaled envelope estimators and separate scaled envelope estimators.