Model selection for density estimation with L2-loss

Probability Theory and Related Fields

Volumn 158, Issue 3-4, 2014, Pages 533-574

  Birgé, Lucien ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Density estimation; Estimator selection; Histograms; L2 loss; Model selection

Indexed keywords

EID: 84897601047     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-013-0488-x     Document Type: Article

References (35)

1. Baraud, Y., Birgé, L.: Estimating composite functions by model selection. http://arxiv. org/abs/1102. 2818. To appear in Ann. Inst. Henri Poincaré Probab. et Statist. (2011).
2. Birgé, L.: Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 181-237 (1983).
3. Birgé, L.: Sur un théorème de minimax et son application aux tests. Probab. Math. Statist. 3, 259-282 (1984).
4. Birgé, L.: Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. et Statist. 42, 273-325 (2006).
5. Birgé, L.: Statistical estimation with model selection. Indag. Math. 17, 497-537 (2006).
6. Birgé, L.: Model selection for Poisson processes. In: Cator, E., Jongbloed, G., Kraaikamp, C., Lopuhaä, R., Wellner, J. (eds.) Asymptotics: particles, processes and inverse problems, Festschrift for Piet Groeneboom. IMS Lect. Notes Monogr. Ser. 55, 32-64 (2007).
7. Birgé, L.: Robust tests for model selection. In: Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., Maathuis, M. H. (eds.) From probability to statistics and back: high-dimensional models and processes - A Festschrift in honor of Jon A. Wellner, vol. 9, pp. 47-64. IMS Collections, Beachwood, Ohio (2013).
8. Birgé, L., Massart, P.: From model selection to adaptive estimation. In: Pollard, D., Torgersen, E., Yang, G. (eds.) Festschrift for Lucien Le Cam: research papers in probability and statistics, pp. 55-87. Springer, New York (1997).
9. Birgé, L., Massart, P.: Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4, 329-375 (1998).
10. Birgé, L., Massart, P.: An adaptive compression algorithm in Besov spaces. Constr. Approx. 16, 1-36 (2000).
11. Birgé, L., Massart, P.: Gaussian model selection. J. Eur. Math. Soc. 3, 203-268 (2001).
12. Birgé, L., Rozenholc, Y.: How many bins should be put in a regular histogram. ESAIM Probab. Stat. 10, 24-45 (2006).
13. Cencov, N. N.: Evaluation of an unknown distribution density from observations. Soviet Math. 3, 1559-1562 (1962).
14. Delyon, B., Juditsky, A.: On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3, 215-228 (1996).
15. Devore, R. A., Lorentz, G. G.: Constructive approximation. Springer, Berlin (1993).
16. Devroye, L.: A course in density estimation. Birkhäuser, Boston (1987).
17. 2 view. Wiley, New York (1985).
18. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., Picard, D.: Density estimation by wavelet thresholding. Ann. Stat. 24, 508-539 (1996).
19. Donoho, D. L., Liu, R. C.: Geometrizing rates of convergence I. Technical report 137. Department of Statistics, University of California, Berkeley (1987).
20. Efromovich, S.: Adaptive estimation and oracle inequalities for probability densities and characteristic functions. Ann. Stat. 36, 1127-1155 (2008).
21. Juditsky, A., Rigollet, P., Tsybakov, A.: Learning by mirror averaging. Ann. Stat. 36, 2183-2206 (2008).
22. Kerkyacharian, G., Picard, D.: Thresholding algorithms, maxisets and well-concentrated bases. Test 9, 283-344 (2000).
23. Le Cam, L. M.: Convergence of estimates under dimensionality restrictions. Ann. Stat. 1, 38-53 (1973).
24. Le Cam, L. M.: On local and global properties in the theory of asymptotic normality of experiments. In: Puri, M. (ed.) Stochastic processes and related topics, vol. 1, pp. 13-54. Academic Press, New York (1975).
25. Le Cam, L. M.: Asymptotic methods in statistical decision theory. Springer, New York (1986).
26. Lounici, K.: Aggregation of density estimators for the Lπ risk with 1≤π<∞. Unpublished manuscript (2008).
27. Massart, P.: Concentration inequalities and model selection. In: Picard, J. (ed.) Lecture on probability theory and statistics, Ecole d'Eté de Probabilités de Saint-Flour XXXIII-2003. Lecture note in mathematics, Springer, Berlin (2007).
28. Pinsker, M. S.: Optimal filtration of square-integrable signals in Gaussian noise. Probl. Inf. Trans. 16, 120-133 (1980).
29. Reynaud-Bouret, P., Rivoirard, V., Tuleau-Malot, C.: Adaptive density estimation: a curse of support? J. Stat. Plann. Inference 141, 115-139 (2011).
30. Rigollet, T.: Ph. D. thesis, University Pierre et Marie Curie, Paris (2006).
31. Rigollet, T., Tsybakov, A. B.: Linear and convex aggregation of density estimators. Math. Methods Stat. 16, 260-280 (2007).
32. Samarov, A., Tsybakov, A. B.: Aggregation of density estimators and dimension reduction. In: Nair, V. (ed.) Advances in statistical modeling and inference. Essays in honor of Kjell A. Doksum, pp. 233-251. World Scientific, Singapore (2007).
33. Yang, Y.: Mixing strategies for density estimation. Ann. Stat. 28, 75-87 (2000).
34. Yang, Y., Barron, A. R.: An asymptotic property of model selection criteria. IEEE Trans. Inf. Theory 44, 95-116 (1998).
35. Yu, B.: Assouad, Fano and Le Cam. In: Pollard, D., Torgersen, E., Yang, G. (eds.) Festschrift for Lucien Le Cam: research papers in probability and statistics, pp. 423-435. Springer, New York (1997).