On Statistical Properties of Sets Fulfilling Rolling-Type Conditions
By Antonio Cuevas, Ricardo Fraiman, Beatriz Pateiro-L'opez. Advances in Applied Probability, 2011
80 citations recorded by the source corpus.
Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain 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 Topsøe'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 (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/2aac2a2e72a24401e5eb103036c9dc5c2396c6d1
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Statistics and Probability
Citations in the corpus
- Estimating the reach of a manifold (2017, background): “The positive reach assumption is routinely imposed in the statistical analysis of geometric structures in order to ensure good statistical properties [15] and to derive theoretical guarantees.”