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The volume-of-tube method for Gaussian random fields with inhomogeneous variance

Author

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  • Kuriki, Satoshi
  • Takemura, Akimichi
  • Taylor, Jonathan E.

Abstract

The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x2100097x
    DOI: 10.1016/j.jmva.2021.104819
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    References listed on IDEAS

    as
    1. 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.
    2. Tamio Koyama & Akimichi Takemura, 2016. "Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables," Computational Statistics, Springer, vol. 31(4), pages 1645-1659, December.
    3. Kato, Naohiro & Kuriki, Satoshi, 2013. "Likelihood ratio tests for positivity in polynomial regressions," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 334-346.
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