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Simultaneous confidence bands for contrasts between several nonlinear regression curves

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  • Lu, Xiaolei
  • Kuriki, Satoshi

Abstract

We propose simultaneous confidence bands of the hyperbolic-type for the contrasts between several nonlinear (curvilinear) regression curves. The critical value of a confidence band is determined from the distribution of the maximum of a chi-square random process defined on the domain of explanatory variables. We use the volume-of-tube method to derive an upper tail probability formula of the maximum of a chi-square random process, which is asymptotically exact and sufficiently accurate in commonly used tail regions. Moreover, we prove that the formula obtained is equivalent to the expectation of the Euler–Poincaré characteristic of the excursion set of the chi-square random process, and hence conservative. This result is therefore a generalization of Naiman’s inequality for Gaussian random processes. As an illustrative example, growth curves of consomic mice are analyzed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:155:y:2017:i:c:p:83-104
    DOI: 10.1016/j.jmva.2016.11.011
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    References listed on IDEAS

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    1. Krivobokova, Tatyana & Kneib, Thomas & Claeskens, Gerda, 2010. "Simultaneous Confidence Bands for Penalized Spline Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 852-863.
    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. 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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    Cited by:

    1. Kuriki, Satoshi & Takemura, Akimichi & Taylor, Jonathan E., 2022. "The volume-of-tube method for Gaussian random fields with inhomogeneous variance," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. Majid Mojirsheibani, 2022. "On the maximal deviation of kernel regression estimators with NMAR response variables," Statistical Papers, Springer, vol. 63(5), pages 1677-1705, October.

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