标题： 关于一致估计维度值 显示英文标题

标题： On consistent estimation of dimension values

摘要： 估计欧几里得空间中紧致子集S的维度的问题，从随机样本点出发进行考虑。 重点放在统计意义上的一致性结果上。 也就是说，当样本量趋向无穷大时，收敛到真实维度值的陈述。 在许多可用的维度定义中，我们侧重于三种概念：Minkowski维度、相关维度以及可能不太流行的点态维度概念。 我们证明了这些量的一些自然估计量的统计一致性。 我们的证明部分依赖于使用一个工具估计量，该估计量以经验体积函数Vn(r)的形式表述，定义为距离样本点的距离最多为r的点集的Lebesgue测度。特别地，我们探讨了目标集S的真实体积函数V(r)在某个从零开始的区间上为多项式的情况。 还包括了一项实证研究。 我们的研究旨在为判断集合S的维度是否小于环境空间的维度这一问题提供一些理论支持和实际见解。 这是维度研究中的一个重要统计动机，与所谓的流形假设密切相关。 显示英文摘要

摘要： The problem of estimating, from a random sample of points, the dimension of a compact subset S of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence to the true dimension value when the sample size grows to infinity. Among the many available definitions of dimension, we have focused (on the grounds of its statistical tractability) on three notions: the Minkowski dimension, the correlation dimension and the, perhaps less popular, concept of pointwise dimension. We prove the statistical consistency of some natural estimators of these quantities. Our proofs partially rely on the use of an instrumental estimator formulated in terms of the empirical volume function Vn (r), defined as the Lebesgue measure of the set of points whose distance to the sample is at most r. In particular, we explore the case in which the true volume function V (r) of the target set S is a polynomial on some interval starting at zero. An empirical study is also included. Our study aims to provide some theoretical support, and some practical insights, for the problem of deciding whether or not the set S has a dimension smaller than that of the ambient space. This is a major statistical motivation of the dimension studies, in connection with the so-called Manifold Hypothesis.