arXiv2025
It is widely known that the tube method, or equivalently the Euler characteristic heuristic, provides a very accurate approximation for the tail probability that the supremum of a smooth Gaussian random field exceeds a threshold value $c$. The relative approximation error $\Delta(c)$ is exponentially small as a function of $c$ when $c$ tends to infinity. On the other hand, little is known about non-Gaussian random fields. In this paper, we obtain the approximation error of the tube method applied to the canonical isotropic random fields on a unit sphere defined by $u\mapsto\langle u,\xi\rangle$, $u\in M\subset\mathbb{S}^{n-1}$, where $\xi$ is a spherically contoured random vector. These random fields have statistical applications in multiple testing and simultaneous regression inference when the unknown variance is estimated. The decay rate of the relative error $\Delta(c)$ depends on the tail of the distribution of $\|\xi\|^2$ and the critical radius of the index set $M$. If this distribution is subexponential but not regularly varying, $\Delta(c)\to 0$ as $c\to\infty$. However, in the regularly varying case, $\Delta(c)$ does not vanish and hence is not negligible. To address this limitation, we provide simple upper and lower bounds for $\Delta(c)$ and for the tube formula itself. Numerical studies are conducted to assess the accuracy of the asymptotic approximation.