### Abstract

A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 - α level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation.

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

Pages (from-to) | 39-57 |

Number of pages | 19 |

Journal | International Statistical Review |

Volume | 76 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2008 |

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### Keywords

- Bivariate normal
- Bivariate t
- Polar coordinators
- Simple linear regression
- Simultaneous inferences

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

**Construction of exact simultaneous confidence bands for a simple linear regression model.** / Liu, Wei; Lin, Shan; Piegorsch, Walter W.

Research output: Contribution to journal › Article

*International Statistical Review*, vol. 76, no. 1, pp. 39-57. https://doi.org/10.1111/j.1751-5823.2007.00027.x

}

TY - JOUR

T1 - Construction of exact simultaneous confidence bands for a simple linear regression model

AU - Liu, Wei

AU - Lin, Shan

AU - Piegorsch, Walter W

PY - 2008/4

Y1 - 2008/4

N2 - A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 - α level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation.

AB - A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 - α level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation.

KW - Bivariate normal

KW - Bivariate t

KW - Polar coordinators

KW - Simple linear regression

KW - Simultaneous inferences

UR - http://www.scopus.com/inward/record.url?scp=41849129536&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41849129536&partnerID=8YFLogxK

U2 - 10.1111/j.1751-5823.2007.00027.x

DO - 10.1111/j.1751-5823.2007.00027.x

M3 - Article

AN - SCOPUS:41849129536

VL - 76

SP - 39

EP - 57

JO - International Statistical Review

JF - International Statistical Review

SN - 0306-7734

IS - 1

ER -