Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-08T21:38:26.152Z Has data issue: false hasContentIssue false
  • English
  • Français

On boundary estimation

Part of: Geometric probability and stochastic geometry Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Antonio Cuevas  and
Alberto Rodríguez-Casal
Antonio Cuevas*
Affiliation:
Universidad Autónoma de Madrid
Alberto Rodríguez-Casal*
Affiliation:
Universidad de Vigo
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain. Email address: [email protected]
∗∗ Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Vigo, Facultad de Ciencias Económicas y Empresariales, 36200 Vigo, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of estimating the boundary of a compact set S ⊂ ℝd from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.

Keywords

Boundary estimationset estimationHausdorff metricexpandable setstandard set

MSC classification

Primary: 62G05: Estimation 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 340 - 354
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baíllo, A. and Cuevas, A. (2001). On the estimation of a star-shaped set. Adv. Appl. Prob. 33, 1–10.Google Scholar
Baíllo, A., Cuevas, A. and Justel, A. (2000). Set estimation and nonparametric detection. Canad. J. Statist. 28, 765–782.CrossRefGoogle Scholar
Bertholet, V., Rasson, J.-P. and Lissoir, S. (1998). About the automatic detection of training sets for multispectral images classification. In Advances in Data Science and Classification, eds Rizzi, A., Vichi, M. and Bock, H. H., Springer, Berlin, pp. 221–226.Google Scholar
Bräker, H., Hsing, T. and Bingham, N. H. (1998). On the Hausdorff distance between a convex set and an interior random convex hull. Adv. Appl. Prob. 30, 295–316.Google Scholar
Carlstein, E. and Krishnamoorthy, C. (1992). Boundary estimation. J. Amer. Statist. Soc. 87, 430–438.CrossRefGoogle Scholar
Cuevas, A. and Fraiman, R. (1997). A plug-in approach to support estimation. Ann. Statist. 25, 2300–2312.Google Scholar
Cuevas, A., Febrero, M. and Fraiman, R. (2000). Estimating the number of clusters. Canad. J. Statist. 28, 367–382.Google Scholar
Cuevas, A., Febrero, M. and Fraiman, R. (2001). Cluster analysis: a further approach based on density estimation. Comput. Statist. Data Anal. 36, 441–459.Google Scholar
Davies, S. and Hall, P. G. (1999). Fractal analysis of surface roughness by using spatial data. J. R. Statist. Soc. B 61, 3–37.CrossRefGoogle Scholar
Devroye, L. and Wise, G. L. (1980). Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38, 480–488.Google Scholar
Donoho, D. (1999). Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27, 859–897.Google Scholar
Dümbgen, L. and Walther, G. (1996). Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 28, 384–393.Google Scholar
Edgar, G. A. (1990). Measure, Topology and Fractal Geometry. Springer, New York.Google Scholar
Edgar, G. A. (1998). Integral, Probability and Fractal Measures. Springer, New York.Google Scholar
Gijbels, I., Mammen, E., Park, B. U. and Simar, L. (1999). On estimation of monotone and concave frontier functions. J. Amer. Statist. Soc. 445, 220–228.Google Scholar
Hall, P. and Rau, C. (2000). Tracking a smooth fault line in a response surface. Ann. Statist. 28, 713–733.Google Scholar
Hall, P., Nussbaum, M. and Stern, S. E. (1997). On the estimation of a support curve of indeterminate sharpness. J. Multivariate Anal. 62, 204–232.CrossRefGoogle Scholar
Hall, P., Park, B. U. and Stern, S. E. (1998). On polynomial estimators of frontiers and boundaries. J. Multivariate Anal. 66, 71–98.CrossRefGoogle Scholar
Härdle, W., Park, B. U. and Tsybakov, A. B. (1995). Estimation of non-sharp support boundaries. J. Multivariate Anal. 55, 205–218.Google Scholar
Janson, S. (1987). Maximal spacings in several dimensions. Ann. Prob. 15, 274–280.Google Scholar
Korostelev, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Springer, New York.Google Scholar
Kumbhakar, S. C. and Lovell, C. A. K. (2003). Stochastic Frontier Analysis. Cambridge University Press.Google Scholar
Marr, D. (1982). Vision. Freeman, San Francisco, CA.Google Scholar
Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press.Google Scholar
Rudemo, M. and Stryhn, H. (1994a). Approximating the distribution of maximum likelihood contour estimators in two-region images. Scand. J. Statist. 21, 41–55.Google Scholar
Rudemo, M. and Stryhn, H. (1994b). Boundary estimation for star-shaped objects. In Change-Point Problems (IMS Lecture Notes Monogr. Ser. 23), eds Carlstein, E., Müller, H.-G. and Siegmund, D., Institute of Mathematical Statistics, Hayward, CA, pp. 276–283.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Tsybakov, A. B. (2004). Optimal aggregation of classifiers in statistical learning. Ann. Statist. 32, 135–166.Google Scholar
Walther, G. (1997). Granulometric smoothing. Ann. Statist. 25, 2273–2299.Google Scholar
Walther, G. (1999). On a generalization of Blaschke's rolling theorem and the smoothing of surfaces. Math. Meth. Appl. Sci. 22, 301–316.Google Scholar
Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.CrossRefGoogle Scholar