On statistical properties of sets fulfilling rolling-type conditions
Entity
UAM. Departamento de MatemáticasPublisher
Applied Probability TrustDate
2012-06Citation
10.1239/aap/1339878713
Advances in Applied Probability 44.2 (2012): 311-329
ISSN
0001-8678 (print); 1475-6064 (online)DOI
10.1239/aap/1339878713Funded by
This work has been partially supported by Spanish Grants MTM2010-17366 and CCG10-UAM/ESP-5494 (A. Cuevas), MTM2010-17366 and Argentinian grant PIC-2008-0921 (R. Fraiman) and by Spanish Grant MTM2008-03010 and the IAP research network grant no. P6/03 from the Belgian government (B. Pateiro-López)Project
Gobierno de España. MTM2010-17366; Gobierno de España. MTM2008-03010Editor's Version
http://dx.doi.org/10.1239/aap/1339878713Subjects
R-convexity; Positive reach; Rolling condition; Glivenko-Cantelli classes; Set estimation; Boundary length; Excess mass; MatemáticasNote
This is the peer reviewed version of the following article: Advances in Applied Probability 44.2 (2012): 311-329, which has been published in final form at http://dx.doi.org/10.1239/aap/1339878713Rights
© 2012 Project EuclidAbstract
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 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 (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)
Files in this item
Google Scholar:Cuevas González, Antonio
-
Fraiman, Ricardo
-
Pateiro-López, Beatriz
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