Tube formula for spherically contoured random fields with subexponential marginals

TL;DR

This paper extends the tube method to non-Gaussian, spherically contoured random fields with subexponential tails, providing bounds on approximation errors and assessing accuracy through numerical studies.

Contribution

It derives the approximation error of the tube method for non-Gaussian fields on spheres with subexponential marginals, including bounds and asymptotic behavior.

Findings

Error $ o 0$ for subexponential tails as threshold increases

Error does not vanish for regularly varying tails

Numerical studies confirm the accuracy of asymptotic approximations

Abstract

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.