### Abstract

In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182-213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.

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

Pages (from-to) | 425-436 |

Number of pages | 12 |

Journal | Journal of Statistical Computation and Simulation |

Volume | 75 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2005 |

Externally published | Yes |

### Fingerprint

### Keywords

- Linear regression
- Sensitivity of a confidence band
- Simultaneous confidence bands
- Statistical simulation

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

*Journal of Statistical Computation and Simulation*,

*75*(6), 425-436. https://doi.org/10.1080/00949650410001701715

**Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals.** / Liu, W.; Jamshidian, M.; Zhang, Ying; Bretz, F.

Research output: Contribution to journal › Article

*Journal of Statistical Computation and Simulation*, vol. 75, no. 6, pp. 425-436. https://doi.org/10.1080/00949650410001701715

}

TY - JOUR

T1 - Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

AU - Liu, W.

AU - Jamshidian, M.

AU - Zhang, Ying

AU - Bretz, F.

PY - 2005/6/1

Y1 - 2005/6/1

N2 - In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182-213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.

AB - In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182-213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.

KW - Linear regression

KW - Sensitivity of a confidence band

KW - Simultaneous confidence bands

KW - Statistical simulation

UR - http://www.scopus.com/inward/record.url?scp=27844573316&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27844573316&partnerID=8YFLogxK

U2 - 10.1080/00949650410001701715

DO - 10.1080/00949650410001701715

M3 - Article

AN - SCOPUS:27844573316

VL - 75

SP - 425

EP - 436

JO - Journal of Statistical Computation and Simulation

JF - Journal of Statistical Computation and Simulation

SN - 0094-9655

IS - 6

ER -