×

zbMATH — the first resource for mathematics

On statistical properties of sets fulfilling rolling-type conditions. (English) Zbl 1252.47089
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 W. Polinik [Ann. Stat. 23, No. 3, 855–881 (1995; Zbl 0841.62045)]).

MSC:
47N30 Applications of operator theory in probability theory and statistics
60D05 Geometric probability and stochastic geometry
62G05 Nonparametric estimation
Software:
alphahull
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Ambrosio, L., Colesanti, A. and Villa, E. (2008). Outer Minkowski content for some classes of closed sets. Math. Ann. 342 , 727-748. · Zbl 1152.28005
[2] Baíllo, A. and Cuevas, A. (2001). On the estimation of a star-shaped set. Adv. Appl. Prob. 33 , 717-726. · Zbl 1003.62030
[3] Biau, G., Cadre, B. and Pelletier, B. (2008). Exact rates in density support estimation. J. Multivariate Anal. 99 , 2185-2207. · Zbl 1151.62027
[4] Biau, G., Cadre, B., Mason, D. M. and Pelletier, B. (2009). Asymptotic normality in density support estimation. Electron. J. Prob. 14 , 2617-2635. · Zbl 1185.62071
[5] Billingsley, P. and Topsøe, F. (1967). Uniformity in weak convergence. Z. Wahrscheinlichkeitsth. 7 , 1-16. · Zbl 0147.15701
[6] Bickel, P. J. and Millar, P. W. (1992). Uniform convergence of probability measures on classes of functions. Statist. Sinica 2 , 1-15. · Zbl 0821.60002
[7] Colesanti, A. and Manselli, P. (2010). Geometric and isoperimetric properties of sets of positive reach in \({\mathbf E}^d\). · Zbl 1242.52008
[8] Cuevas, A. and Fraiman, R. (2009). Set estimation. In New Perspectives on Stochastic Geometry, eds W. S. Kendall and I. Molchanov, Oxford University Press, pp. 366-389.
[9] Cuevas, A. and Rodríguez-Casal, A. (2004). On boundary estimation. Adv. Appl. Prob. 36 , 340-354. · Zbl 1045.62019
[10] Cuevas, A., Fraiman, R. and Rodríguez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35 , 1031-1051. · Zbl 1124.62017
[11] Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York. · Zbl 0853.68150
[12] Dümbgen, L. and Walther, G. (1996). Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 28 , 384-393. · Zbl 0861.60022
[13] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 , 418-491. · Zbl 0089.38402
[14] Gruber, P. M. and Wills, J. M. (eds) (1993). Handbook of Convex Geometry. North-Holland, Amsterdam. · Zbl 0777.52001
[15] Hartigan, J. A. (1987). Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 82 , 267-270. · Zbl 0607.62045
[16] Jiménez, R. and Yukich, J. E. (2011). Nonparametric estimation of surface integrals. Ann. Statist. 39 , 232-260. · Zbl 1209.62059
[17] Mani-Levitska, P. (1993) Characterizations of convex sets. In Handbook of Convex Geometry , eds P. M. Gruber and J. M. Wills, North-Holland, Amsterdam, pp. 19-41. · Zbl 0847.52001
[18] Mason, D. M. and Polonik, W. (2009). Asymptotic normality of plug-in level set estimates. Ann. Appl. Prob. 19 , 1108-1142. · Zbl 1180.62048
[19] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces . Cambridge University Press. · Zbl 0819.28004
[20] Müller, D. W. and Sawitzki, G. (1991). Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 , 738-746. · Zbl 0733.62040
[21] Pateiro-López, B. and Rodríguez-Casal, A. (2008). Length and surface area estimation under smoothness restrictions. Adv. Appl. Prob. 40 , 348-358. · Zbl 1416.62201
[22] Pateiro-López, B. and Rodríguez-Casal, A. (2010). Generalizing the convex hull of a sample: the R package alphahull. J. Statist. Software 34 , 1-28.
[23] Perkal, J. (1956). Sur les ensembles \(\varepsilon\)-convexes. Colloq. Math. 4 , 1-10. · Zbl 0071.38101
[24] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[25] Polonik, W. (1995). Measuring mass concentrations and estimating density contour clusters–an excess mass approach. Ann. Statist. 23 , 855-881. · Zbl 0841.62045
[26] Polonik, W. and Wang, Z. (2005). Estimation of regression contour clusters–an application of the excess mass approach to regression. J. Multivariate Anal. 94 , 227-249. · Zbl 1066.62047
[27] Rataj, J. (2005). On boundaries of unions of sets with positive reach. Beiträge Algebra Geom. 46 , 397-404. · Zbl 1097.53050
[28] Reitzner, M. (2009). Random polytopes. In New Perspectives on Stochastic Geometry , eds W. S. Kendall and I. Molchanov, Oxford University Press, pp. 45-75.
[29] Rodríguez-Casal, A. (2007). Set estimation under convexity-type assumptions. Ann. Inst. H. Poincaré Prob. Statist. 43 , 763-774. · Zbl 1169.62317
[30] Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. · Zbl 0910.62001
[31] Walther, G. (1997). Granulometric smoothing. Ann. Statist. 25 , 2273-2299. · Zbl 0919.62026
[32] Walther, G. (1999). On a generalization of Blaschke’s rolling theorem and the smoothing. Math. Methods Appl. Sci. 22 , 301-316. · Zbl 0933.52003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.