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.

翻译：管法或体积管法接近一个平滑高斯随机字段的最大尾部概率,无平均值和单位差异。该方法评估指数集的球形管体积,然后将其转换为尾部概率。在本研究中,我们将管法概括到一个差异不常数的情况。我们以曲度强力提供球形管体体积公式,以及高斯随机场最大不相容的尾部概率公式及其拉比近差。特别是,该管的临界半径是用来评价无湿度近似误的。举例来说,我们讨论Wishart 矩阵体中最大的eigen值分布的近似值,并使用非特性矩阵参数。当指数集是定数时,Bonferroni 法是管管子法的定数。我们为Bonferroni 方法在不常值差时提供公式,用于Bonferroni 方法的负值近似误差。