### Abstract

A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set optimality for confidence bands in simple linear regression, J. Amer. Statist. Assoc. 102 (477) (2007) pp. 181-190] proposed the use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can be generalized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and hence the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and hence the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band. Crown

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

Pages (from-to) | 1432-1439 |

Number of pages | 8 |

Journal | Journal of Multivariate Analysis |

Volume | 100 |

Issue number | 7 |

DOIs | |

State | Published - Aug 2009 |

### Fingerprint

### Keywords

- Confidence sets
- Linear regression
- Simultaneous confidence bands
- Statistical inference

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Numerical Analysis
- Statistics and Probability

### Cite this

*Journal of Multivariate Analysis*,

*100*(7), 1432-1439. https://doi.org/10.1016/j.jmva.2008.12.003

**Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion.** / Liu, W.; Hayter, A. J.; Piegorsch, Walter W; Ah-Kine, P.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 100, no. 7, pp. 1432-1439. https://doi.org/10.1016/j.jmva.2008.12.003

}

TY - JOUR

T1 - Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion

AU - Liu, W.

AU - Hayter, A. J.

AU - Piegorsch, Walter W

AU - Ah-Kine, P.

PY - 2009/8

Y1 - 2009/8

N2 - A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set optimality for confidence bands in simple linear regression, J. Amer. Statist. Assoc. 102 (477) (2007) pp. 181-190] proposed the use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can be generalized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and hence the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and hence the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band. Crown

AB - A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set optimality for confidence bands in simple linear regression, J. Amer. Statist. Assoc. 102 (477) (2007) pp. 181-190] proposed the use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can be generalized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and hence the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and hence the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band. Crown

KW - Confidence sets

KW - Linear regression

KW - Simultaneous confidence bands

KW - Statistical inference

UR - http://www.scopus.com/inward/record.url?scp=64249157647&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64249157647&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2008.12.003

DO - 10.1016/j.jmva.2008.12.003

M3 - Article

AN - SCOPUS:64249157647

VL - 100

SP - 1432

EP - 1439

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 7

ER -