### Abstract

We propose a new parsimonious version of the classical multivariate normal linear model, yielding a maximum likelihood estimator (MLE) that is asymptotically less variable than the MLE based on the usual model. Our approach is based on the construction of a link between the mean function and the covariance matrix, using the minimal reducing subspace of the latter that accommodates the former. This leads to a multivariate regression model that we call the envelope model, where the number of parameters is maximally reduced. The MLE from the envelope model can be substantially less variable than the usual MLE, especially when the mean function varies in directions that are orthogonal to the directions of maximum variation for the covariance matrix.

Original language | English (US) |
---|---|

Pages (from-to) | 927-960 |

Number of pages | 34 |

Journal | Statistica Sinica |

Volume | 20 |

Issue number | 3 |

State | Published - Jul 1 2010 |

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### Keywords

- Discriminant analysis
- Functional data analysis
- Grassmann manifolds
- Invariant subspaces
- Principal components
- Reduced rank regression
- Reducing subspaces
- Sufficient dimension reduction

### Cite this

*Statistica Sinica*,

*20*(3), 927-960.

**Envelope models for parsimonious and efficient multivariate linear regression.** / Cook, R. D; Li, Bing; Chiaromonte, Francesca.

Research output: Contribution to journal › Article

*Statistica Sinica*, vol. 20, no. 3, pp. 927-960.

}

TY - JOUR

T1 - Envelope models for parsimonious and efficient multivariate linear regression

AU - Cook, R. D

AU - Li, Bing

AU - Chiaromonte, Francesca

PY - 2010/7/1

Y1 - 2010/7/1

N2 - We propose a new parsimonious version of the classical multivariate normal linear model, yielding a maximum likelihood estimator (MLE) that is asymptotically less variable than the MLE based on the usual model. Our approach is based on the construction of a link between the mean function and the covariance matrix, using the minimal reducing subspace of the latter that accommodates the former. This leads to a multivariate regression model that we call the envelope model, where the number of parameters is maximally reduced. The MLE from the envelope model can be substantially less variable than the usual MLE, especially when the mean function varies in directions that are orthogonal to the directions of maximum variation for the covariance matrix.

AB - We propose a new parsimonious version of the classical multivariate normal linear model, yielding a maximum likelihood estimator (MLE) that is asymptotically less variable than the MLE based on the usual model. Our approach is based on the construction of a link between the mean function and the covariance matrix, using the minimal reducing subspace of the latter that accommodates the former. This leads to a multivariate regression model that we call the envelope model, where the number of parameters is maximally reduced. The MLE from the envelope model can be substantially less variable than the usual MLE, especially when the mean function varies in directions that are orthogonal to the directions of maximum variation for the covariance matrix.

KW - Discriminant analysis

KW - Functional data analysis

KW - Grassmann manifolds

KW - Invariant subspaces

KW - Principal components

KW - Reduced rank regression

KW - Reducing subspaces

KW - Sufficient dimension reduction

UR - http://www.scopus.com/inward/record.url?scp=78349232766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78349232766&partnerID=8YFLogxK

M3 - Article

VL - 20

SP - 927

EP - 960

JO - Statistica Sinica

JF - Statistica Sinica

SN - 1017-0405

IS - 3

ER -