Abstract
A family of confidence bands (simultaneous confidence regions) is given for EY = x′β that are piecewise-linear in x. Normality is assumed. These confidence bands are advocated over the usual hyperbolic band when the region of prime interest is distant from \({\overline{\bf x}}\). In particular, this is the case when x = x(t) for time t and future time is of primary interest, that is for the prediction problem. For the case x′ = (1, t), the family of bands includes that of Graybill and Bowden (J Am Stat Assoc 62:403–408, 1967).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Saidy OM, Piegorsch WW, West RW, Nitcheva DK (2003) Confidence bands for low-dose risk estimation with quantal response data. Biometrics 59: 1056–1062
Bebbington M, Lai CD, Zitikis R (2010) Life expectancy of a bathtub shaped failure distribution. Stat Papers 51: 599–612
Bowden DC (1970) Simultaneous confidence bands for linear regression models. J Am Stat Assoc 65: 413–421
Chen ZM (1998) Joint estimation for the parameters of the extreme value distributions. Stat Papers 39: 135–146
Graybill FA, Bowden DC (1967) Linear segment confidence bands for simple linear models. J Am Stat Assoc 62: 403–408
Grob J (2003) Linear regression. Lecture notes in statistics, vol 175. Springer, Berlin
Hayter AJ, Liu W, Ah-Kine P (2009) A ray method of confidence band construction for multiple linear regression models. J Stat Plan Infer 139: 329–334
Kahle W (1994) Simultaneous confidence-regions for the parameters of damage processes. Stat Papers 35: 27–41
Liu W, Hayter AJ (2007) Minimum area confidence set optimality for confidence bands in simple linear regression. J Am Stat Assoc 102: 181–190
Liu W, Hayter AJ, Piegorsch WW, Ah-Kine P (2009) Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion. J Multivar Anal 100: 1432–1439
Liu W, Jamshidian M, Zhang Y (2004) Multiple comparison of several regression models. J Am Stat Assoc 99: 395–403
Liu W, Jamshidian M, Zhang Y, Bretz F (2005a) Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals. J Stat Comput Simul 75: 425–436
Liu W, Jamshidian M, Zhang Y, Donnelly J (2005b) Simulation-based simultaneous confidence bands for a multiple linear regression model when the covariates are constrained. J Comput Graph Stat 14: 459–484
Liu W, Lin S (2009) Construction of exact simultaneous confidence bands in multiple linear regression with predictor variables constrained in an ellipsoidal region. Statistica Sinica 19: 213–232
Liu W, Lin S, Piegorsch WW (2008) Construction of exact simultaneous confidence bands for a simple linear regression model. International Statistical Review 76: 39–57
Montgomery DC, Peck EA, Vining GG (2001) Introduction to linear regression analysis, 3rd edn. John Wiley and Sons, New York
Piegorsch WW, West RW, Pan W, Kodell R (2005) Low dose risk estimation via simultaneous statistical inferences. J R Stat Soc C 54: 245–258
Scheffé H (1953) A method for judging all contrasts in analysis of variance. Biometrika 40: 87–104
Seber GAF (1977) Linear regression analysis. Wiley, New York
Seber GAF, Lee AJ (2003) Linear regression analysis, 2nd edn. John Wiley and Sons, Hoboken
Weisberg S (2005) Applied linear regression, 3rd edn. John Wiley and Sons, Hoboken
Working H, Hotelling H (1929) Applications of the theory of error to the interpretation of trends. J Am Stat Assoc 24: 73–85
Yan X, Su XG (2009) Linear regression analysis: theory and computing. World Scientific Publishing Company, Hackensack
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Withers, C.S., Nadarajah, S. Maximum modulus confidence bands. Stat Papers 53, 811–819 (2012). https://doi.org/10.1007/s00362-011-0384-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-011-0384-3