Abstract: Inference based on the partial envelope model is variational or nonequivariant under rescaling of the responses, and tends to restrict its use to responses measured in identical or analogous units. The efficiency acquisitions promised by partial envelopes frequently cannot be accomplished when the responses are measured in diverse scales. Here, we extend the partial envelope model to a scaled partial envelope model that overcomes the aforementioned disadvantage and enlarges the scope of partial envelopes. The proposed model maintains the potential of the partial envelope model in terms of efficiency and is invariable to scale changes. Further, we demonstrate the maximum likelihood estimators and their properties. Lastly, simulation studies and a real-data example demonstrate the advantages of the scaled partial envelope estimators, including a comparison with the standard model estimators, partial envelope estimators, and scaled envelope estimators.

Keywords: Dimension reduction, grassmannian, scaled envelope model, partial envelope model, scale invariance.