Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. Relevant results in this field, such as rates of convergence and confidence sets, depend on the standardness parameter, which in practice may be unknown. In this paper, we review the notion of standardness and its connection to other geometrical restrictions. We prove the almost sure consistency of a plug-in type estimator for the so-called standardness constant, already studied in the literature. We propose a method to correct the bias of the plug-in estimator and corroborate our theoretical findings through a small simulation study. We also show that it is not possible to determine, based on a finite sample, whether a probability measure satisfies the standard assumption.

翻译：标准性是集合估计文献中广泛采用的假设，也出现在拓扑数据分析的统计方法中——该领域通常假设数据采样自满足标准性假设的概率测度。相关结论（如收敛速率与置信集）依赖于实践中可能未知的标准性参数。本文回顾了标准性的概念及其与其他几何约束的关联，证明了文献中已研究的所谓标准性常数的插入型估计量的几乎处处相合性，并提出一种校正插入型估计量偏差的方法，通过小规模模拟研究验证了理论结果。此外，我们证明无法基于有限样本判断概率测度是否满足标准性假设。