Statistical analysis of measures of non-convexity

TL;DR

This paper analyzes various measures of non-convexity for sets, introduces a new measure, and studies their statistical properties, including consistency and asymptotic distribution, with practical applications demonstrated on real data.

Contribution

It introduces a new measure of non-convexity and provides statistical analysis, including consistency and asymptotic distribution, for existing and new measures based on sample data.

Findings

Proved consistency of estimators for non-convexity measures

Derived asymptotic distributions for the estimators

Demonstrated practical applicability on real data

Abstract

Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set $S$, by dealing with their estimation based on a sample of points in $S$. We introduce also a new measure of non-convexity. We discuss briefly about these different notions of non-convexity, prove consistency and find the asymptotic distribution for the proposed estimators. We also consider the practical implementation of these estimators and illustrate their applicability to a real data example.