  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.
