The volume-of-tube method for Gaussian random fields with inhomogeneous variance.
The volume-of-tube method for Gaussian random fields with inhomogeneous variance.
复制标题
非齐次方差高斯随机场的管体积法。
DOI：
10.1016/j.jmva.2021.104819
复制
发表时间：
2022
影响因子：
1.6
通讯作者：
A. Takemura and Jonathan E. Taylor.
中科院分区：
文献类型：
--
作者：
Satoshi Kuriki;A. Takemura and Jonathan E. Taylor.
管法或管体积法近似具有零均值和单位方差的平滑高斯随机场最大值的尾部概率。该方法评估围绕指标集的球形管的体积，然后将其转换为尾部概率。在本研究中，我们将管法推广到方差不恒定的情况。我们以曲率张量的形式给出了非恒定半径球管的体积公式，以及非齐次方差高斯随机场最大值的尾部概率公式及其拉普拉斯近似。特别是，管的临界半径被推广用于评估渐近逼近误差。作为一个例子，我们讨论具有非单位矩阵参数的 Wishart 矩阵的最大特征值分布的近似。 Bonferroni方法是指标集为有限集时的管方法。我们提供了当方差不恒定时 Bonferroni 方法的渐近逼近误差的公式。
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.