Set estimation: An overview and some recent developments

Recent Advances and Trends in Nonparametric Statistics

Volumn , Issue , 2003, Pages 251-264

  Cuevas, Antonio ^a   Rodríguez Casal, Alberto ^b  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^b UNIVERSITY OF VIGO   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33644675986     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/B978-044451378-6/50017-X     Document Type: Chapter

References (57)

1. p version of the Hausdorff metric. Nieuw Arch. Wisk. 1992, 4:157-183.
2. Baíllo A., Cuevas A. On the estimation of a star-shaped set. Adv. Appl. Probab. 2001, 33:1-10.
3. Baíllo A., Cuesta-Albertos J.A., Cuevas A. Convergence rates in nonparametric estimation of level sets. Statist. Probab. Lett. 2001, 53:27-35.
4. Baíllo A., Cuevas A., Justel A. Set estimation and nonparametric detection. Canad. J. Statist. 2000, 28:765-782.
5. Banker R.D. Maximum likelihood, consistency and data envelopment analysis: a statistical foundation. Management Science 1993, 39:1265-1273.
6. Candès E.J., Donoho D.L. New tight frames of curvelets and optimal representations of objects with smooth singularities, preprint 2002, Stanford University, New York.
7. Carlstein E., Krishnamoorty C. Boundary estimation. J. Amer. Statist. Assoc. 1992, 87:430-438.
8. Charnes A., Cooper W., Rhodes E. Measuring the inefficiency of decision making units. European J. Oper. Res. 1978, 6:429-444.
9. Chaudhury A.R., Basu A., Bhandari S.K., Chaudhury B.B. An efficient approach to consistent set estimation. Sankhya Ser. B 1999, 61:496-513.
10. Chevalier J. Estimation du support et du contour de support d'une loi de probabilité. Ann. Inst. H. Poincaré 1976, 12:339-364. sec. B.
11. Cuevas A. On pattern analysis in the nonconvex case. Kybernetes 1990, 19:26-33.
12. Cuevas A., Febrero M., Fraiman R. Estimating the number of clusters. Canad. J. Statist. 2000, 28:367-382.
13. Cuevas A., Febrero M., Fraiman R. Cluster analysis: a further approach based on density estimation. Comp. Stat. Data Analysis 2001, 36:441-459.
14. Cuevas A., Fraiman A., Fraiman R. A plug-in approach to support estimation. Ann. Statist. 1997, 25:2300-2312.
15. A. Cuevas and A. Rodríguez-Casal, On boundary estimation, preprint, Univ. Autónoma de Madrid and Univ. de Vigo.
16. Devroye L., Wise G.L. Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 1980, 38:480-488.
17. Devroye L., Györfi L., Lugosi G. A Probabilisty Theory of Pattern Recognition 1996, Springer-Verlag.
18. Donoho D.L. Wedgelets: nearly minimax estimation of edges. Ann. Statist. 1999, 27:859-897.
19. Dümbgen L., Walther G. Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 1996, 28:384-393.
20. Edgar G.A. Measure, Topology and Fractal Geometry 1990, Springer-Verlag, New York.
21. Farrell M.J. The measurement of productive efficiency. J. Roy. Statist. Soc. Ser. A 1957, 120:253-281.
22. Gijbels I., Mammen E., Park B.U., Simar L. On estimation of monotones and concave frontier funtions. J. Amer. Statist. Assoc. 1999, 94:220-228.
23. Grenander U. Abstract Inference 1980, Wiley, New York.
24. Hall P., Nussbaum M., Stern S.E. On the estimation of a support curve of indeterminate sharness. J. Multivariate Anal. 1997, 62:204-232.
25. Hall P., Peng L., Rau C. Local likelihood tracking of fault lines and boundaries. J. R. Stat. Soc. Ser. B 2001, 63:569-582.
26. Hall P., Park B.U., Stern S.E. On polynomial estimators of frontiers and boundaries. J. Multivariate Anal. 1998, 66:71-98.
27. Hall P., Rau C. Tracking a smooth fault line in a response surface. Ann. Statist. 2000, 28:713-733.
28. Hall P., Rau C. Likelihood-based confidence bands for fault lines in response surfaces. Probab. Theory Related Fields 2002, 124:26-49.
29. Härdle W., Park B.U., Tsybakov A.B. Estimation of non-sharp support boundaries. J. Multivariate Anal. 1995, 55:205-218.
30. Hartigan J.A. Clustering Algorithms 1975, Wiley, New York.
31. Hartigan J.A. Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 1987, 82:267-270.
32. Hastie T., Tibshirani T., Tibshirani R., Friedman R., Friedman J. The Elements of Statistical Learning 2001, Springer-Verlag, New York.
33. Janson S. Maximal spacings in several dimensions. Ann. Probab. 1987, 15:274-280.
34. Kneip A., Park B.U., Simar L. A note on the convergence of nonparametric DEA efficiency measures. Econometric Theory 1998, 14:783-793.
35. Korostelev A., Simar L., Tsybakov A.B. Efficient estimation of monotone boundaries. Ann. Statist. 1995, 23:476-489.
36. Korostelev A., Simar L., Tsybakov A.B. On estimation of monotone and convex boundaries. Publ. Inst. Statist. Univ. Paris 1995, 39:3-18.
37. Korostelev A.P., Tsybakov A.B. Minimax Theory of Image Reconstruction, Lecture Notes in Statistics 82 1993, Springer-Verlag, New York.
38. Mammen E., Tsybakov A.B. Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 1995, 23:502-524.
39. Matheron G. Random Sets and Integral Geometry 1975, Wiley, New York.
40. Molchanov I.S. A limit theorem for solutions of inequalities. Scand. J. Statist. 1998, 25:235-242.
41. Müller D.W., Sawitzki G. Excess mass estimates and tests of multimodality. J. Amer. Statist. Assoc. 1991, 86:738-746.
42. Nolan D. The excess-mass ellipsoid. J. Multivariate Anal. 1991, 39:348-371.
43. Polonik W. Measuring mass concentration and estimating density contour clusters-an excess mass approach. Ann. Statist. 1995, 23:855-881.
44. Polonik W. Concentration and goodness-of-fit in higher dimensions: (Asymptotically) distribution-free methods. Ann. Statist. 1999, 27:1210-1229.
45. Rényi A., Sulanke A., Sulanke R. Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrs. und Verw. Gebiete 1963, 2:75-84.
46. Rényi A., Sulanke A., Sulanke R. Über die konvexe Hülle von n zufällig gewählten Punkten (II). Z. Wahrs. und Verw. Gebiete 1964, 2:138-147.
47. Ripley B.D., Rasson J.P. Finding the edge of a Poisson forest. J. Appl. Prob. 1977, 14.
48. A. Rodríguez-Casal, Estimacón de conjuntos y sus fronteras. Un enfoque geométrico. Ph. D. Thesis, Univ. Santiago de Compostela.
49. Rudemo M., Stryhn H. Approximating the distribution of maximum likelihood contour estimators in two-region images. Scand. J. Statist. 1994, 21:41-55.
50. Schneider R. Random approximation of convex sets. J. Microscopy 1988, 151:211-227.
51. Serra J. Image Analysis and Mathematical Morphology 1982, Academic Press, New York.
52. Tsybakov A.B. On nonparametric estimation of density level sets. Ann. Statist. 1997, 25:948-969.
53. Tsybakov A.B. Optimal aggregation of classifiers in statistical learning 2001, preprint, Université Paris VI.
54. Walther G. Granulometric smoothing. Ann. of Statist. 1997, 25:2273-2299.
55. Walther G. On a generalization of Blaschke's Rolling Theorem and the smoothing of surfaces. Math. Meth. Appl. Sci. 1999, 22:301-316.
56. Wand M.P., Jones M.C. Kernel Smoothing 1995, Chapman and Hall, London.
57. Weil W., Wieacker J.A. Stochastic Geometry. Handbook of convex geometry 1993, Vol. A, B:1391-1438. North-Holland, London.