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

Alejandro Cholaquidis, Leonardo Moreno, Beatriz Pateiro-López

TL;DR

This paper investigates standardness, a geometric-probabilistic regularity condition in set estimation, by defining the standardness constant $\Upsilon(S,\nu)$ and studying its estimability from i.i.d. samples. It analyzes a plug-in estimator $\hat{\Upsilon}_n$ and proves its almost-sure consistency under broad assumptions, but shows poor finite-sample performance, motivating a bias-corrected estimator $\tilde{\Upsilon}_n$ with guaranteed almost-sure convergence. Through simulations in 2D and higher dimensions (including uniform and nonuniform sampling), the authors demonstrate that $\tilde{\Upsilon}_n$ substantially improves accuracy and stability over $\hat{\Upsilon}_n$. A key theoretical contribution is the non-estimability result: even when the standardness constant can be consistently estimated, one cannot, from finite data, decide whether the support is standard with respect to the sampling measure, highlighting intrinsic limits in distributional inference for geometric functionals.

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.

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

TL;DR

This paper investigates standardness, a geometric-probabilistic regularity condition in set estimation, by defining the standardness constant and studying its estimability from i.i.d. samples. It analyzes a plug-in estimator and proves its almost-sure consistency under broad assumptions, but shows poor finite-sample performance, motivating a bias-corrected estimator with guaranteed almost-sure convergence. Through simulations in 2D and higher dimensions (including uniform and nonuniform sampling), the authors demonstrate that substantially improves accuracy and stability over . A key theoretical contribution is the non-estimability result: even when the standardness constant can be consistently estimated, one cannot, from finite data, decide whether the support is standard with respect to the sampling measure, highlighting intrinsic limits in distributional inference for geometric functionals.

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.
Paper Structure (16 sections, 11 theorems, 69 equations, 1 figure, 3 tables)

This paper contains 16 sections, 11 theorems, 69 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Some notation
  4. Shape constraints beyond convexity
  5. Standardness in set estimation
  6. The standardness constant
  7. Estimation of the standardness constant
  8. A bias-correction method
  9. Simulations
  10. Uniform samples on R2
  11. Uniform samples on Rd for d>2
  12. Non-uniform samples
  13. Non-decidability
  14. Concluding remarks
  15. Proofs of the theoretical results in Sections \ref{['sec-back']} and \ref{['sec:est_stand']}
  16. ...and 1 more sections

Key Result

Lemma 3.2

Assume that the support $S\subset\mathbb{R}^d$ of a probability measure $\nu$ fulfils condition (H1). Then, $S$ is standard with respect to $\nu$ and where $\Upsilon(S,\nu)$ is the standardness constant defined by stconst.

Figures (1)

  • Figure 1: The sets shown in (a) and (b) are standard with respect to $\mu$. The set in (c) is not standard with respect to $\mu$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Remark 1
  • Lemma 3.3
  • Proposition 1
  • Corollary 1
  • Corollary 2
  • Theorem 4.1
  • ...and 16 more