标题： 关于维度值的相合估计 显示英文标题

标题： On consistent estimation of dimension values

摘要： 从随机样本点中估计欧几里得空间中紧子集$S$的维数的问题被考虑。 重点放在统计意义上的相容性结果。 也就是说，当样本量趋于无穷时，收敛到真实维数值的陈述。 在众多维数定义中，我们（基于其统计可处理性）专注于三个概念：闵可夫斯基维数、关联维数以及可能不太流行的点态维数。 我们证明了这些量的一些自然估计量的统计相容性。 我们的证明部分依赖于使用一个用经验体积函数$V_n(r)$表述的工具性估计量，该函数定义为距离样本不超过$r$的点的集合的勒贝格测度。 特别是，我们探讨了目标集$S$的真实体积函数$V(r)$在从零开始的某个区间上是多项式的情况。 还包含了一个实证研究。 我们的研究旨在为判断集合$S$的维数是否小于环境空间的维数的问题提供一些理论支持和实际见解。 这是与所谓的“流形假设”相关的维度研究的主要统计动机。 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 $V_n(r)$, 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''.