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Simultaneous Envelopes for Multivariate Linear Regression

Summary: This article introduces envelopes for simultaneously reducing the predictors and the responses in multivariate linear regression, so the regression then depends only on estimated linear combinations of X and Y. The authors use a likelihood-based objective function for estimating envelopes and then propose algorithms for estimation of a simultaneous envelope as well as for basic Grassmann manifold optimization. The asymptotic properties of the resulting estimator are studied under normality and extended to general distributions. The authors also investigate likelihood ratio tests and information criteria for determining the simultaneous envelope dimensions. Simulation studies and real data examples show substantial gain over the classical methods, like partial least squares, canonical correlation analysis, and reduced-rank regression. This article has supplementary material available online.

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  • Topics: Statistics
  • Keywords: Multivariate analysis, Linear regression, Canonical analysis, Least squares, Principal components
  • Author: Cook, R. Dennis; Zhang, Xin;
  • Journal: Technometrics