On standardness and the non-estimability of certain functionals of a set

TL;DR

This paper examines the concept of standardness in set estimation and topological data analysis, proving the consistency of estimators for the standardness constant, proposing bias correction, and demonstrating the non-estimability of standardness from finite samples.

Contribution

It introduces a consistent estimator for the standardness constant, proposes a bias correction method, and proves the non-estimability of standardness from finite data.

Findings

Proved almost sure consistency of the standardness constant estimator

Developed a bias correction method for the estimator

Showed non-estimability of standardness from finite samples

Abstract

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.