论文标题
$ U $ -MAX统计信息的广义限制定理
Generalized Limit Theorems For $U$-max Statistics
论文作者
论文摘要
Lao和Mayer在2008年引入了U-MAX统计数据。他们认为原始样品的所有可能子集对内核的平均，而是考虑了核的最大值。这种统计在随机几何形状上是自然的。例子是由圆，椭圆等随机点形成的多边形和多面体的最大周围和区域。研究研究限制U-Max统计定理的主要方法是泊松近似。在本文中，我们考虑了在圆上定义的一般内核类，我们证明了具有Weibull分布的通用限制定理是一个限制。它的参数取决于内核的程度，其最大点的结构和内核的Hessians在这些点上。到目前为止，几乎所有已知的限制定理都可以作为我们一般定理的简单特殊案例获得。我们还考虑了几个新示例。此外，我们不仅考虑点的均匀分布，而且还要在满足轻度附加条件的圆上几乎是任意分布。
U-max statistics were introduced by Lao and Mayer in 2008. Instead of averaging the kernel over all possible subsets of the original sample, they considered the maximum of the kernel. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for U-max statistics is a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.
