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Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model

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  • Wei Liu
  • Shan Lin
  • Walter W. Piegorsch

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. Un intervalle de confiance simultanée fournit une variété d'inférences sur les composantes inconnues d'un modéle de régression. Plusieurs articles récents utilisent des intervalles de confiance dans des buts variés; voir par exemple Sun, Raz et Faraway (1999), Spurrier (1999), Al‐Saidy et al. (2003), Liu, Jamshidian et Zhang (2004), Bhargava et Spurrier (2004), Piegorsch et al. (2005), Liu et al. (2007). La construction d'intervalles de confiance simultanés pour un simple modéle de régression linéaire a une histoire riche, qui remonte aux travaux de Working et hotelling (1929). L'objet de cet article est de consolider la littérature moderne disparate sur les intervalles de confiance simultanés dans la régression linéaire, de fournir des expressions pour la construction d'intervalles de confiance simultanés de niveau exact 1 −α pour un modéle de régression linéaire simple ou pour des formes unilatérales ou bilatérales. Nous concentrons notre attention sur les trois formes les plus reconnues: hyperbolique, à deux segments et à trois segments (qui est aussi appelée forme trapézoïdale et inclut un intervalle de largeur constante comme cas spécial). Certaines de ces expressions sont déjà apparues dans la littérature statistique, d'autres sont nouvellement introduites dans cet article. Les dérivations comprennent typiquement un vecteur aléatoire standard bivarié t et sa transformation en coordonnées polaires.

Suggested Citation

  • Wei Liu & Shan Lin & Walter W. Piegorsch, 2008. "Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model," International Statistical Review, International Statistical Institute, vol. 76(1), pages 39-57, April.
  • Handle: RePEc:bla:istatr:v:76:y:2008:i:1:p:39-57
    DOI: 10.1111/j.1751-5823.2007.00027.x
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    1. Liu, W. & Hayter, A.J. & Piegorsch, W.W. & Ah-Kine, P., 2009. "Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1432-1439, August.
    2. Rand R. Wilcox, 2017. "Linear regression: robust heteroscedastic confidence bands that have some specified simultaneous probability coverage," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(14), pages 2564-2574, October.
    3. Francq, Bernard G. & Govaerts, Bernadette, 2012. "Hyperbolic confidence bands of errors-in-variables regression lines applied to method comparison studies," LIDAM Discussion Papers ISBA 2012039, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Lu, Xiaolei & Kuriki, Satoshi, 2017. "Simultaneous confidence bands for contrasts between several nonlinear regression curves," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 83-104.
    5. Christopher Withers & Saralees Nadarajah, 2012. "Maximum modulus confidence bands," Statistical Papers, Springer, vol. 53(4), pages 811-819, November.
    6. Jamshidian, Mortaza & Liu, Wei & Bretz, Frank, 2010. "Simultaneous confidence bands for all contrasts of three or more simple linear regression models over an interval," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1475-1483, June.

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