### Abstract

This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.

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

Pages (from-to) | 459-484 |

Number of pages | 26 |

Journal | Journal of Computational and Graphical Statistics |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2005 |

Externally published | Yes |

### Fingerprint

### Keywords

- Inequality constraints
- Linear regression
- Polyhedral cone
- Projection
- Quadratic programming
- Statistical simulation

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Computational and Graphical Statistics*,

*14*(2), 459-484. https://doi.org/10.1198/106186005X47408

**Simulation-based simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals.** / Liu, W.; Jamshidian, M.; Zhang, Ying; Donnelly, J.

Research output: Contribution to journal › Article

*Journal of Computational and Graphical Statistics*, vol. 14, no. 2, pp. 459-484. https://doi.org/10.1198/106186005X47408

}

TY - JOUR

T1 - Simulation-based simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

AU - Liu, W.

AU - Jamshidian, M.

AU - Zhang, Ying

AU - Donnelly, J.

PY - 2005/6/1

Y1 - 2005/6/1

N2 - This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.

AB - This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.

KW - Inequality constraints

KW - Linear regression

KW - Polyhedral cone

KW - Projection

KW - Quadratic programming

KW - Statistical simulation

UR - http://www.scopus.com/inward/record.url?scp=13444255042&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=13444255042&partnerID=8YFLogxK

U2 - 10.1198/106186005X47408

DO - 10.1198/106186005X47408

M3 - Article

VL - 14

SP - 459

EP - 484

JO - Journal of Computational and Graphical Statistics

JF - Journal of Computational and Graphical Statistics

SN - 1061-8600

IS - 2

ER -