Tube formula for spherically contoured random fields with subexponential marginals

TL;DR

This work analyzes the tube method and Euler characteristic heuristic for non-Gaussian, spherically contoured random fields on the sphere, focusing on the tail probability of the field's supremum over a manifold $M$. By classifying the tail of the radial part $R_n=\|\xi\|^2$ into a broad family $\mathcal{L}_{\beta,\gamma}$, the authors derive precise asymptotics for the relative error $\Delta(c)$ between the tube approximation and the true excursion probability, showing tail-validity for light-tailed and subexponential tails but not for regularly varying tails. The key driver is the critical radius $\theta_*$ of $M$, which determines the rate and even the existence of the leading-term corrections, with explicit formulas involving Beta-type random variables and geometric integrals. The results yield practical Bonferroni-type bounds and guidance for simultaneous inference when the marginal variance is estimated, highlighting scenarios (e.g., regularly varying tails) where the tube method may fail and where bounds are essential. Overall, the paper clarifies when the tube method provides reliable tail approximations beyond Gaussian fields and offers quantitative tools for applied statistical problems such as multiple testing and simultaneous confidence bands.

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 $Δ(c)$ 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 $u\mapsto\langle u,ξ\rangle$, $u\in M\subset\mathbb{S}^{n-1}$, where $ξ$ 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 $Δ(c)$ depends on the tail of the distribution of $\|ξ\|^2$ and the critical radius of the index set $M$. If this distribution is subexponential but not regularly varying, $Δ(c)\to 0$ as $c\to\infty$. However, in the regularly varying case, $Δ(c)$ does not vanish and hence is not negligible. To address this limitation, we provide simple upper and lower bounds for $Δ(c)$ and for the tube formula itself. Numerical studies are conducted to assess the accuracy of the asymptotic approximation.

Figures (5)

• Figure 1: Voronoi cells $D_i$ on the unit sphere, distance $\theta(u,v)$ and its minimum $\min_v\theta(u,v)$.
• Figure 2: Values of $\log\mathbb{P}(T_{\max}>c)$ (left) and $\Delta(c)$ (right) for the $t_3$-distributed random field $(T_i)_{i=1,2,3}$.
• Figure 3: Values of $\log\mathbb{P}(T_{\max}>c)$ (left) and $\Delta(c)$ (right) for the lognormal-distributed random field $(T_i)_{i=1,2,3}$.
• Figure 4: Values of $\log\mathbb{P}(T_{\max}>c)$ (left) and $\Delta(c)$ (right) for the Bessel-distributed random field $(T_i)_{i=1,2,3}$.
• Figure 5: Values of $\log\mathbb{P}(T_{\max}>c)$ (left) and $\Delta(c)$ (right) for the Gaussian random field $(T_i)_{i=1,2,3}$.

Theorems & Definitions (31)

• Proposition 1
• Remark 1
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 2
• proof
• Proposition 3