On statistical properties of sets fulfilling rolling-type conditions

TL;DR

This paper explores the relationships between shape conditions like positive reach, r-convexity, and rolling conditions, providing new consistency results for set estimation methods and their boundary properties.

Contribution

It introduces new theoretical insights into the relations among shape conditions and establishes full consistency of estimators under these conditions.

Findings

Full consistency for set estimation under rolling conditions

The class of sets with reach ≥ r is a P-uniformity and Glivenko-Cantelli class

Convergence of boundary length of the r-convex hull to the true support boundary

Abstract

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we obtain for the estimation of sets fulfilling a rolling condition a result of "full consistency" (i.e., consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsoe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (a.s.) to that of the underlying support. Fifth, the above results are applied to get new consistency statements for level set estimators based on the excess mass methodology (Polonik, 1995).