@article{AIHPB_2006__42_3_273_0, author = {Birge, Lucien}, title = {Model selection via testing : an alternative to (penalized) maximum likelihood estimators}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, volume = {42}, year = {2006}, pages = {273-325}, doi = {10.1016/j.anihpb.2005.04.004}, mrnumber = {2219712}, zbl = {05024238}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPB_2006__42_3_273_0} }
Birgé, Lucien. Model selection via testing : an alternative to (penalized) maximum likelihood estimators. Annales de l'I.H.P. Probabilités et statistiques, Tome 42 (2006) pp. 273-325. doi : 10.1016/j.anihpb.2005.04.004. http://gdmltest.u-ga.fr/item/AIHPB_2006__42_3_273_0/
[1] Deux remarques sur l'estimation, C. R. Acad. Sci. Paris, Sér. I Math. 296 (1983) 1021-1024. | MR 777600 | Zbl 0568.62003
,[2] J.-Y. Audibert, Théorie statistique de l'apprentissage : une approche PAC-bayésienne, Thèse de doctorat, Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, Paris, 2004.
[3] Model selection for regression on a random design, ESAIM Probab. Statist. 6 (2002) 127-146. | Numdam | MR 1918295 | Zbl 1059.62038
,[4] Complexity regularization with applications to artificial neural networks, in: (Ed.), Nonparametric Functional Estimation, Kluwer, Dordrecht, 1991, pp. 561-576. | MR 1154352 | Zbl 0739.62001
,[5] Risk bounds for model selection via penalization, Probab. Theory Related Fields 113 (1999) 301-415. | MR 1679028 | Zbl 0946.62036
, , ,[6] Minimum complexity density estimation, IEEE Trans. Inform. Theory 37 (1991) 1034-1054. | MR 1111806 | Zbl 0743.62003
, ,[7] On the asymptotic normality of the -error in partitioning regression estimation, J. Statist. Plann. Inference 71 (1998) 93-107. | MR 1651863 | Zbl 0961.62030
, ,[8] Approximation dans les espaces métriques et théorie de l'estimation, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 65 (1983) 181-237. | MR 722129 | Zbl 0506.62026
,[9] Sur un théorème de minimax et son application aux tests, Probab. Math. Statist. 3 (1984) 259-282. | MR 764150 | Zbl 0571.62036
,[10] Stabilité et instabilité du risque minimax pour des variables indépendantes équidistribuées, Ann. Inst. H. Poincaré Sect. B 20 (1984) 201-223. | Numdam | MR 762855 | Zbl 0542.62018
,[11] On estimating a density using Hellinger distance and some other strange facts, Probab. Theory Related Fields 71 (1986) 271-291. | MR 816706 | Zbl 0561.62029
,[12] Model selection for Gaussian regression with random design, Bernoulli 10 (2004) 1039-1051. | MR 2108042 | Zbl 1064.62030
,[13] Rates of convergence for minimum contrast estimators, Probab. Theory Related Fields 97 (1993) 113-150. | MR 1240719 | Zbl 0805.62037
, ,[14] From model selection to adaptive estimation, in: , , (Eds.), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, Springer-Verlag, New York, 1997, pp. 55-87. | MR 1462939 | Zbl 0920.62042
, ,[15] Minimum contrast estimators on sieves: exponential bounds and rates of convergence, Bernoulli 4 (1998) 329-375. | MR 1653272 | Zbl 0954.62033
, ,[16] An adaptive compression algorithm in Besov spaces, Constr. Approx. 16 (2000) 1-36. | MR 1848840 | Zbl 1004.41006
, ,[17] Gaussian model selection, J. Eur. Math. Soc. 3 (2001) 203-268. | MR 1848946 | Zbl 1037.62001
, ,[18] Piecewise-polynomial approximation of functions of the classes , Mat. Sb. 73 (1967) 295-317. | MR 217487 | Zbl 0173.16001
, ,[19] Asymptotic equivalence of nonparametric regression and white noise, Ann. Statist. 24 (1996) 2384-2398. | MR 1425958 | Zbl 0867.62022
, ,[20] F. Bunea, A.B. Tsybakov, M.H. Wegkamp, Aggregation for regression learning, Technical report 948, Laboratoire de Probabilités, Université Paris VI, 2004, http://www.proba.jussieu.fr/mathdoc/preprints/index.html# 2004.
[21] G. Castellan, Modified Akaike's criterion for histogram density estimation, Technical report 99.61, Université Paris-Sud, Orsay, 1999, http://www.math.u-psud.fr/~biblio/pub/1999/.
[22] Sélection d'histogrammes à l'aide d'un critère de type Akaike, C. R. Acad. Sci. Paris 330 (2000) 729-732. | MR 1763919 | Zbl 0969.62023
,[23] O. Catoni, The mixture approach to universal model selection, Technical report LMENS-97-22, Ecole Normale Supérieure, Paris, 1997, http://www.dma.ens.fr/edition/publis/1997/titre97.html. | Zbl 0928.62033
[24] Statistical learning theory and stochastic optimization, in: (Ed.), Lecture on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXXI - 2001, Lecture Note in Math., vol. 1851, Springer-Verlag, Berlin, 2004. | MR 2163920 | Zbl 1076.93002
,[25] A measure of asymptotic efficiency of tests of a hypothesis based on a sum of observations, Ann. Math. Statist. 23 (1952) 493-507. | MR 57518 | Zbl 0048.11804
,[26] R.A. DeVore, G. Kerkyacharian, D. Picard, V. Temlyakov, Mathematical methods for supervised learning, Technical report 0422, IMI, University of South Carolina, Columbia, 2004, http://www.math.sc.edu/imip/preprints/04.html.
[27] Constructive Approximation, Springer-Verlag, Berlin, 1993. | MR 1261635 | Zbl 0797.41016
, ,[28] Combinatorial Methods in Density Estimation, Springer-Verlag, New York, 2001. | MR 1843146 | Zbl 0964.62025
, ,[29] Density estimation by wavelet thresholding, Ann. Statist. 24 (1996) 508-539. | MR 1394974 | Zbl 0860.62032
, , , ,[30] Minimax risk over hyperrectangles, and implications, Ann. Statist. 18 (1990) 1416-1437. | MR 1062717 | Zbl 0705.62018
, , ,[31] Maximum Penalized Likelihood Estimation, vol. I: Density Estimation, Springer, New York, 2001. | MR 1837879 | Zbl 0984.62026
, ,[32] Some current developments in density estimation, in: , , (Eds.), Mathematics and Computer Science, CWI Monograph, vol. 1, Elsevier, Amsterdam, 1986, pp. 163-192. | MR 873578 | Zbl 0593.62030
,[33] A Distribution-Free Theory of Nonparametric Regression, Springer, New York, 2002. | Zbl 1021.62024
, , , ,[34] A robust version of the probability ratio test, Ann. Math. Statist. 36 (1965) 1753-1758. | MR 185747 | Zbl 0137.12702
,[35] Robust Statistics, John Wiley, New York, 1981. | MR 606374 | Zbl 0536.62025
,[36] Chi-square oracle inequalities, in: , , (Eds.), State of the Art in Probability and Statistics, Festschrift for Willem R. van Zwet, Lecture Notes Monograph Ser., vol. 36, Institute of Mathematical Statistics, 2001, pp. 399-418. | MR 1836572
,[37] Functional aggregation for nonparametric estimation, Ann. Statist. 28 (2000) 681-712. | MR 1792783 | Zbl 1105.62338
, ,[38] Thresholding algorithms, maxisets and well-concentrated bases, Test 9 (2000) 283-344. | MR 1821645 | Zbl 1107.62323
, ,[39] ε-entropy and ε-capacity of sets in function spaces, Amer. Math. Soc. Transl. (2) 17 (1961) 277-364. | Zbl 0133.06703
, ,[40] Adaptive estimation of a quadratic functional by model selection, Ann. Statist. 28 (2000) 1302-1338. | MR 1805785 | Zbl 1105.62328
, ,[41] On the assumptions used to prove asymptotic normality of maximum likelihood estimates, Ann. Math. Statist. 41 (1970) 802-828. | MR 267676 | Zbl 0246.62039
,[42] L.M. Le Cam, Limits of experiments, in: Proc. 6th Berkeley Symp. on Math. Stat. and Prob. I, 1972, pp. 245-261. | MR 415819 | Zbl 0271.62004
[43] Convergence of estimates under dimensionality restrictions, Ann. Statist. 1 (1973) 38-53. | MR 334381 | Zbl 0255.62006
,[44] On local and global properties in the theory of asymptotic normality of experiments, in: (Ed.), Stochastic Processes and Related Topics, vol. 1, Academic Press, New York, 1975, pp. 13-54. | MR 395005 | Zbl 0389.62011
,[45] Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986. | MR 856411 | Zbl 0605.62002
,[46] Maximum likelihood: an introduction, Inter. Statist. Rev. 58 (1990) 153-171. | Zbl 0715.62045
,[47] Metric dimension and statistical estimation, CRM Proc. and Lecture Notes 11 (1997) 303-311. | MR 1479680 | Zbl 0942.62035
,[48] Approximation of Functions, Holt, Rinehart, Winston, New York, 1966. | MR 213785 | Zbl 0153.38901
,[49] Constructive Approximation, Advanced Problems, Springer, Berlin, 1996. | MR 1393437 | Zbl 0910.41001
, , ,[50] Topics in non-parametric statistics, in: (Ed.), Lecture on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXVIII - 1998, Lecture Notes in Math., vol. 1738, Springer-Verlag, Berlin, 2000, pp. 85-297. | MR 1775640 | Zbl 0998.62033
,[51] Asymptotic equivalence of density estimation and Gaussian white noise, Ann. Statist. 24 (1996) 2399-2430. | MR 1425959 | Zbl 0867.62035
,[52] n-widths in Approximation Theory, Springer-Verlag, Berlin, 1985. | MR 774404 | Zbl 0551.41001
,[53] Optimal filtration of square-integrable signals in Gaussian noise, Problems Inform. Transmission 16 (1980) 120-133. | MR 624591 | Zbl 0452.94003
,[54] Convergence rates of sieve estimates, Ann. Statist. 22 (1994) 580-615. | MR 1292531 | Zbl 0805.62008
, ,[55] On the estimation of a probability density function by the maximum penalized likelihood method, Ann. Statist. 10 (1982) 795-810. | MR 663433 | Zbl 0492.62034
,[56] Optimal rates of aggregation, in: Proceedings of 16th Annual Conference on Learning Theory (COLT) and 7th Annual Workshop on Kernel Machines, Lecture Notes in Artificial Intelligence, vol. 2777, Springer-Verlag, Berlin, 2003, pp. 303-313.
,[57] Estimating a regression function, Ann. Statist. 18 (1990) 907-924. | MR 1056343 | Zbl 0709.62040
,[58] Hellinger-consistency of certain nonparametric maximum likelihood estimates, Ann. Statist. 21 (1993) 14-44. | MR 1212164 | Zbl 0779.62033
,[59] Empirical Processes in M-Estimation, Cambridge University Press, Cambridge, 2000. | MR 1739079
,[60] Asymptotic Statistics, Cambridge University Press, Cambridge, 1998. | MR 1652247 | Zbl 0910.62001
,[61] Spline Models for Observational Data, SIAM, Philadelphia, PA, 1990. | MR 1045442 | Zbl 0813.62001
,[62] Note on the consistency of the maximum likelihood estimate, Ann. Math. Statist. 20 (1949) 595-601. | MR 32169 | Zbl 0034.22902
,[63] Model selection in nonparametric regression, Ann. Statist. 31 (2003) 252-273. | MR 1962506 | Zbl 1019.62037
,[64] Probability inequalities for likelihood ratios and convergence rates of sieve MLEs, Ann. Statist. 23 (1995) 339-362. | MR 1332570 | Zbl 0829.62002
, ,[65] Y. Yang, Minimax optimal density estimation, Ph.D. dissertation, Dept. of Statistics, Yale University, New Haven, 1996.
[66] Mixing strategies for density estimation, Ann. Statist. 28 (2000) 75-87. | MR 1762904 | Zbl 1106.62322
,[67] Combining different procedures for adaptive regression, J. Multivariate Anal. 74 (2000) 135-161. | MR 1790617 | Zbl 0964.62032
,[68] Adaptive regression by mixing, J. Amer. Statist. Assoc. 96 (2001) 574-588. | MR 1946426 | Zbl 1018.62033
,[69] Y. Yang, How accurate can any regression procedure be?, Technical report, Iowa State University, Ames, 2001, http://www.public.iastate.edu/yyang/papers/index.html.
[70] Aggregating regression procedures to improve performance, Bernoulli 10 (2004) 25-47. | MR 2044592 | Zbl 1040.62030
,[71] An asymptotic property of model selection criteria, IEEE Trans. Inform. Theory 44 (1998) 95-116. | MR 1486651 | Zbl 0949.62041
, ,[72] Information-theoretic determination of minimax rates of convergence, Ann. Statist. 27 (1999) 1564-1599. | MR 1742500 | Zbl 0978.62008
, ,[73] Rates of convergence of minimum distance estimates and Kolmogorov's entropy, Ann. Statist. 13 (1985) 768-774. | MR 790571 | Zbl 0576.62057
,[74] Assouad, Fano and Le Cam, in: , , (Eds.), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, Springer-Verlag, New York, 1997, pp. 423-435. | MR 1462963 | Zbl 0896.62032
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