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

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  • Liu, W.
  • Hayter, A.J.
  • Piegorsch, W.W.
  • Ah-Kine, P.

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.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:7:p:1432-1439
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    References listed on IDEAS

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    1. Liu, W. & Hayter, A.J., 2007. "Minimum Area Confidence Set Optimality for Confidence Bands in Simple Linear Regression," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 181-190, March.
    2. Liu W. & Jamshidian M. & Zhang Y., 2004. "Multiple Comparison of Several Linear Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 395-403, January.
    3. 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.
    4. Obaid M. Al-Saidy & Walter W. Piegorsch & R. Webster West & Daniela K. Nitcheva, 2003. "Confidence Bands for Low-Dose Risk Estimation with Quantal Response Data," Biometrics, The International Biometric Society, vol. 59(4), pages 1056-1062, December.
    5. Walter W. Piegorsch & R. Webster West & Wei Pan & Ralph L. Kodell, 2005. "Low dose risk estimation via simultaneous statistical inferences," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(1), pages 245-258, January.
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    Cited by:

    1. Christopher Withers & Saralees Nadarajah, 2012. "Maximum modulus confidence bands," Statistical Papers, Springer, vol. 53(4), pages 811-819, November.
    2. Freyberger, Joachim & Rai, Yoshiyasu, 2018. "Uniform confidence bands: Characterization and optimality," Journal of Econometrics, Elsevier, vol. 204(1), pages 119-130.

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