Smooth discrimination analysis

Annals of Statistics

Volumn 27, Issue 6, 1999, Pages 1808-1829

  Mammen, Enno ^a   Tsybakov, Alexandre B ^b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Bayes risk; Discrimination analysis; Empirical risk; Optimal rates; Sieves

Indexed keywords

EID: 0033234630     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1017939240     Document Type: Article

References (28)

1. ALEXANDER, K. S. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12 1041-1067. [Correction: (1987) 15 428-430.]
2. ASSOUAD, P. (1983). Deux remarques sur l'estimation. C. R. Acad. Sci. Paris 296 1021-1024.
3. BARRON, A. (1991). Complexity regularization with application to artificial neural networks. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 561-576. Kluwer, Dordrecht.
4. BARRON, A. (1994). Approximation and estimation bounds for artificial neural networks. Machine Learning 14 115-133.
5. BARRON, A., BIRGÉ, L. and MASSART, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413.
6. BIRGÉ, L. and MASSART, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150.
7. BIRGÉ, L. and MASSART, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 329-375.
8. BLOCK, D. A. and SILVERMAN, B. W. (1997). Monotone discriminant functions and their applications in rheumathology. J. Amer. Statist. Assoc. 92 144-153.
9. BRETAGNOLLE, J. and HUBER, C. (1979). Estimation des densités: risque minimax. Z. Warsch. Verw. Gebiete 47 119-137.
10. DEVROYE, L., GYÖRFI, L. and LUGOSI, G. (1996). A Probabilistic Theory of Pattern Recognition. Springer, New York.
11. DUDLEY, R. M. (1974). Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10 227-236.
12. HARTIGAN, J. A. (1987). Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 82 267-270.
13. KOROSTELEV, A. P. and TSYBAKOV, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, New York.
14. MAMMEN, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741-759.
15. MAMMEN, E. and TSYBAKOV, A. B. (1995). Asymptotic minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502-524.
16. MARRON, J. S. (1983). Optimal rates of convergence to Bayes risk in nonparametric discrimination. Ann. Statist. 11 1142-1155.
17. MÜLLER, D. W. (1993). The excess mass approach in statistics. Beiträge zur Statistik 3. Inst. Math, für Angewandte, Univ. Heidelberg.
18. MÜLLER, D. W. (1995). A backward-induction algorithm for computing the best convex contrast of two bivariate samples. Beiträge zur Satistik 29. Inst. für Angewandte, Univ. Heidelberg.
19. MÜLLER, D. W. and SAWITZKI, G. (1991). Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 738-746.
20. POLONIK, W. (1995). Measuring mass concentrations and estimating density contour clusters: an excess mass approach. Ann. Statist. 23 855-881.
21. RUDEMO, M. and STRYHN, H. (1994). Approximating the distributions of maximum likelihood contour estimates in two-region images. Scand. J. Statist. 21 41-56.
22. TSYBAKOV, A. B. (1997). On nonparametric estimation of density level sets. Ann. Statist. 25 948-969.
23. VAN DE GEER, S. (1991). The entropy bound for monotone functions. Technical Report 91-100. Univ. Leiden.
24. VAN DE GEER, S. (1995). The method of sieves and minimum contrast estimates. Math. Methods Statist. 4 20-38.
25. VAN DE GEER, S. (1998). Applications of Empirical Process Theory to M-estimation. Unpublished manuscript.
26. VAPNIK, V. N. (1996). The Nature of Statistical Learning Theory. Springer, New York.
27. VAPNIK, V. N. and CHERVONENKIS, A. JA. (1974). Theory of Pattern Recognition. Nauka, Moscow (in Russian).
28. WONG, W. H. and SHEN, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362.