On statistical properties of sets fulfilling rolling-type conditions (Q2898907)

Positive reach, \(r\)-convexity and rolling shape conditions for compact sets are studied and motivated by set estimation. The main result is that 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. This is an interesting contribution to the theory of nonparametric boundary estimation which so far relies mostly on the use of two samples (one inside and the other outside the set \(S\)). The efficiency of the results is demonstrated to get new consistency statements for level set estimators based on the excess mass methodology (readers may refer here to \textit{W. Polinik} [Ann. Stat. 23, No. 3, 855--881 (1995; Zbl 0841.62045)]).