infinite dimensional空间中分布的最深点
The deepest point for distributions in infinite dimensional spaces
摘要：
数据云中心的识别是统计学中的基本问题之一。这种中心的一个流行的选择是中位数，并且文献中已经研究了有限维空间中的中位数的几个版本。特别是，基于不同数据深度概念的中位数已经被许多研究人员广泛研究，他们将中位数定义为深度函数达到其最大值的点。换句话说，根据该定义，中值是样本空间中的最深点。本文研究无穷维空间中概率分布的最深点。我们证明了在函数空间中，对于一些著名的深度函数，如带深和半区深，对于许多著名的和常用的概率模型，可能不存在任何有意义的最深点.另一方面，这些深度函数以及空间深度函数的某些修改版本，可以在任何希尔伯特空间中定义，导致一些有用的概念的最深点具有良好的几何和统计特性。这些最深点的经验版本可以方便地计算功能数据，我们证明了这一点，使用一些模拟和真实的数据集。
英文摘要：
Identification of the center of a data cloud is one of the basic problems in statistics. One popular choice for such a center is the median, and several versions of median in finite dimensional spaces have been studied in the literature. In particular, medians based on different notions of data depth have been extensively studied by many researchers, who defined median as the point, where the depth function attains its maximum value. In other words, the median is the deepest point in the sample space according to that definition. In this paper, we investigate the deepest point for probability distributions in infinite dimensional spaces. We show that for some well-known depth functions like the band depth and the half-region depth in function spaces, there may not be any meaningful deepest point for many well-known and commonly used probability models. On the other hand, certain modified versions of those depth functions as well as the spatial depth function, which can be defined in any Hilbert space, lead to some useful notions of the deepest point with nice geometric and statistical properties. The empirical versions of those deepest points can be conveniently computed for functional data, and we demonstrate this using some simulated and real data sets.
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