标题： 球面轮廓随机场的管公式与次指数边缘分布 显示英文标题

标题： Tube formula for spherically contoured random fields with subexponential marginals

摘要： 众所周知，管方法或等价的欧拉特征启发式方法对于平滑高斯随机场的上确界超过阈值$c$的尾部概率提供了非常精确的近似。 相对近似误差$\Delta(c)$作为$c$的函数，当$c$趋向于无穷大时是指数级小的。 另一方面，关于非高斯随机场了解甚少。 在本文中，我们得到了将管方法应用于单位球面上由$u\mapsto\langle u,\xi\rangle$，$u\in M\subset\mathbb{S}^{n-1}$定义的标准各向同性随机场的近似误差，其中$\xi$是一个球面轮廓随机向量。 这些随机场在未知方差被估计时，在多重检验和同时回归推断中有统计应用。 相对误差$\Delta(c)$的衰减率取决于$\|\xi\|^2$分布的尾部和索引集$M$的临界半径。 如果该分布是次指数但不是规则变化的，$\Delta(c)\to 0$随$c\to\infty$而变化。 然而，在规则变化的情况下，$\Delta(c)$不会消失，因此不可忽略。 为了解决这一限制，我们提供了$\Delta(c)$以及管公式本身的简单上下界。 进行了数值研究以评估渐近近似的准确性。 显示英文摘要

摘要： 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.