Inner envelopes: Efficient estimation in multivariate linear regression

Zhihua Su, R. Dennis Cook

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


In this article we propose a new model, called the inner envelope model, which leads to efficient estimation in the context of multivariate normal linear regression. The asymptotic distribution and the consistency of its maximum likelihood estimators are established. Theoretical results, simulation studies and examples all show that the efficiency gains can be substantial relative to standard methods and to the maximum likelihood estimators from the envelope model introduced recently by Cook et al. (2010). Compared to the envelope model, the inner envelope model is based on a different construction and it can produce substantial efficiency gains in situations where the envelope model offers no gains. In effect, inner envelopes open a new frontier to the way in which reducing subspaces can be used to improve efficiency in multivariate problems.

Original languageEnglish (US)
Pages (from-to)687-702
Number of pages16
Issue number3
StatePublished - Sep 1 2012


  • Dimension reduction
  • Envelope model
  • Grassmann manifold
  • Reducing subspace

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