Scaled envelopes: Scale-invariant and efficient estimation in multivariate linear regression

Efficient estimation of the regression coefficients is a fundamental problem in multivariate linear regression. The envelope model proposed by Cook et al. (2010) was shown to have the potential to achieve substantial efficiency gains by accounting for linear combinations of the response vector that are essentially immaterial to coefficient estimation. This requires in part that the distribution of those linear combinations be invariant to changes in the nonstochastic predictor vector. However, inference based on an envelope is not invariant or equivariant under rescaling of the responses, tending to limit application to responses that are measured in the same or similar units. The efficiency gains promised by envelopes often cannot be realized when the responses are measured in different scales. To overcome this limitation and broaden the scope of envelope methods, we propose a scaled version of the envelope model, which preserves the potential of the original envelope methods to increase efficiency and is invariant to scale changes. Likelihood-based estimators are derived and theoretical properties of the estimators are studied in various circumstances. It is shown that estimating appropriate scales for the responses can produce substantial efficiency gains when the original envelope model offers none. Simulations and an example are given to support the theoretical claims.