Some applications of concentration inequalities to statistics

Pascal Massart

Annales de la Faculté des sciences de Toulouse : Mathématiques (2000)

  • Volume: 9, Issue: 2, page 245-303
  • ISSN: 0240-2963

How to cite

top

Massart, Pascal. "Some applications of concentration inequalities to statistics." Annales de la Faculté des sciences de Toulouse : Mathématiques 9.2 (2000): 245-303. <http://eudml.org/doc/73516>.

@article{Massart2000,
author = {Massart, Pascal},
journal = {Annales de la Faculté des sciences de Toulouse : Mathématiques},
keywords = {chi-square statistics; penalty functions; contrast functions; bounded regression; Poisson inequality; Han's inequality; entropy; histogram},
language = {eng},
number = {2},
pages = {245-303},
publisher = {UNIVERSITE PAUL SABATIER},
title = {Some applications of concentration inequalities to statistics},
url = {http://eudml.org/doc/73516},
volume = {9},
year = {2000},
}

TY - JOUR
AU - Massart, Pascal
TI - Some applications of concentration inequalities to statistics
JO - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY - 2000
PB - UNIVERSITE PAUL SABATIER
VL - 9
IS - 2
SP - 245
EP - 303
LA - eng
KW - chi-square statistics; penalty functions; contrast functions; bounded regression; Poisson inequality; Han's inequality; entropy; histogram
UR - http://eudml.org/doc/73516
ER -

References

top
  1. [1] Akaike ( H.). — Information theory and an extension of the maximum likelihood principle. In P.N. Petrov and F. Csaki, editors, Proceedings 2nd International Symposium on Information Theory, pages 267-281. Akademia Kiado, Budapest, 1973. Zbl0283.62006MR483125
  2. [2] Baraud ( Y.). — Model selection for regression on a fixed design. Technical report #97.49, (1997) Université Paris-Sud (to appear in Probability Theory and Related Fields). Zbl0997.62027MR1777129
  3. [3] Baraud ( Y.), Comte ( F.) and Viennet ( G.) (1999). — Model selection for (auto)regression with dependent data. Technical Report LMENS 99-12. Ecole Normale Supérieure, Paris. 
  4. [4] Bennett ( G.). — Probability inequalities for the sum of independent random variables. Journal of the American Statistical Association5733-45 (1962). Zbl0104.11905
  5. [5] Bahadur ( R.R.). - Examples of inconsistency of maximum likelihood estimates. Sankhya Ser.A20, 207-210 (1958). Zbl0087.34202MR107331
  6. [6] Barron ( A.R.) and Sheu ( C.H.). — Approximation of density functions by sequences of exponential families. Ann. Statist., 19:1054-1347, 1991. Zbl0739.62027MR1126328
  7. [7] Barron ( A.R.), Birgé ( L.), Massart ( P.). — Risk bounds for model selection via penalization. Probab. Th. Rel. Fields.113, 301-415 (1999). Zbl0946.62036MR1679028
  8. [8] Birgé ( L.) and Massart ( P.). — Rates of convergence for minimum contrast estimators. Probab. Th. Relat. Fields97, 113-150 (1993). Zbl0805.62037MR1240719
  9. [9] Birgé ( L.) and Massart ( P.). — Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli, 4(3), 329-375 (1998). Zbl0954.62033MR1653272
  10. [10] Birgé ( L.) and Massart ( P.). - From model selection to adaptive estimation. In Festschrift for Lucien Lecam: Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. Yang, eds.), 55-87 (1997) Springer-Verlag, New-York. Zbl0920.62042MR1462939
  11. [11] Birgé ( L.) and Massart ( P.). — Gaussian model selection. Technical report (1999). 
  12. [12] Birgé ( L.) and Massart ( P.). — A generalized cross-validation criterion for density estimation. Unpublished manuscript. 
  13. [13] Birgé ( L.) and Rozenholc ( Y.). — How many bins should be put in a regular histogram. Unpublished manuscript. Zbl1136.62329
  14. [14] Bobkov ( S.). — On Gross' and Talagrand's inequalities on the discrete cube. Vestnik of Syktyvkar University Ser. 1,112-19 (1995) (in Russian). Zbl1011.60500MR1717705
  15. [15] Borell ( C.). — The Brunn-Minkowski inequality in Gauss space. Invent. Math.30, 207-216 (1975). Zbl0292.60004MR399402
  16. [16] Boucheron ( S.), Lugosi ( G.) and Massart ( P.). — A sharp concentration inequality with applications. Technical report # 99.25 (1999), Université de Paris-Sud. MR1749290
  17. [17] CIREL'SON ( B.S.), Ibragimov ( I.A.) and Sudakov ( V.N.). — Norm of Gaussian sample function. In Proceedings of the 3rd Japan-U.S.S.R. Symposium on Probability Theory, Lecture Notes in Mathematics55020-41 (1976) Springer-Verlag, Berlin. Zbl0359.60019MR458556
  18. [18] CIREL'SON ( B.S.) and Sudakov ( V.N.). - Extremal properties of half spaces for spherically invariant measures. J. Soviet. Math.9, 9-18 (1978); translated from Zap. Nauch. Sem. L.O.M.I.41, 14-24 (1974). Zbl0395.28007MR365680
  19. [19] Cover ( T.M.) and Thomas ( J.A.). - Elements of Information Theory. Wiley series in telecommunications. Wiley (1991). Zbl0762.94001MR1122806
  20. [20] Dembo ( A.). — Information inequalities and concentration of measure. Ann. Prob.25927-939 (1997). Zbl0880.60018MR1434131
  21. [21] Donoho ( D.L.) and Johnstone ( I.M.). — Ideal spatial adaptation by wavelet shrinkage. Biometrika81, 425-455 (1994) Zbl0815.62019MR1311089
  22. [22] Gross ( L.) Logarithmic Sobolev inequalities. Amer. J. Math.971061-1083 (1975). Zbl0318.46049MR420249
  23. [23] Castellan ( G.). - Modified Akaike's criterion for histogram density estimation. Technical report #99.61, (1999) Université de Paris-Sud. 
  24. [24] Castellan ( G.). — Density estimation via exponential model selection. Technical report (1999). 
  25. [25] Hoeffding ( W.). — Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association5813-30 (1963). Zbl0127.10602MR144363
  26. [26] Korostelev ( A.P.) and Tsybakov ( A.B.). — Minimax theory of of image reconstruction. Lectures notes in Statistics82, Soringer Verlag, NewYork (1993). Zbl0833.62039MR1226450
  27. [27] Ledoux ( M.). — Isoperimetry and Gaussian Analysis. In Probabilités de St-Flour XXIV-1994 (P. Bernard, ed.), 165-294 (1996) Springer, Berlin. Zbl0874.60005MR1600888
  28. [28] Ledoux ( M.). — On Talagrand deviation inequalities for product measures. ESAIM: Probability and Statistics1, 63-87 (1996) http://www.emath.fr/ps/. Zbl0869.60013MR1399224
  29. [29] Ledoux ( M.) and Talagrand ( M.). — Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete (1991) Springer-Verlag. Zbl0748.60004MR1102015
  30. [30] Marton ( K.). — A simple proof of the blowing up lemma. IEEE Trans. Inform. Theory IT-32445-446 (1986). Zbl0594.94003MR838213
  31. [31] Marton ( K.). — Bounding d-distance by information divergence: a method to prove measure concentration. Ann. Prob.24927-939 (1996). Zbl0865.60017MR1404531
  32. [32] Mason ( D.M.) and van Zwet ( W.R.) A refinement of the KMT inequality for the uniform empirical process. Ann. Prob.15, 871-884 (1987). Zbl0638.60040MR893903
  33. [33] Massart ( P.). — About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Prob. (To appear)(2000). Zbl1140.60310MR1782276
  34. [34] Massart ( P.). — Optimal constants for Hoeffding type inequalities. Technical report (1998). 
  35. [35] Mcdiarmid ( C.). — On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148-188. Cambridge University Press, Cambridge, 1989. Zbl0712.05012MR1036755
  36. [36] Talagrand ( M.). — An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities in product spaces. Proc. Amer. Math. Soc.104905-909 (1988). Zbl0691.60015MR964871
  37. [37] Talagrand ( M.). — Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l'I.H.E.S.8173-205 (1995). Zbl0864.60013MR1361756
  38. [38] Talagrand ( M.). — Sharper bounds for empirical processes. Annals of Probability22, 28-76 (1994). Zbl0798.60051MR1258865
  39. [39] Talagrand ( M.). — New concentration inequalities in product spaces. Invent. Math.126, 505-563 (1996). Zbl0893.60001MR1419006
  40. [40] Vapnik ( V.N.). — Estimation of dependencies based on empirical data. Springer, New York. Zbl0499.62005
  41. [41] Vapnik ( V.N.). — Statistical learning theory. J. Wiley, New York. Zbl0935.62007MR1641250
  42. [42] Van Der Vaart ( A.). — Asymptotic statistics. Cambridge University Press (1998). Zbl0910.62001MR1652247
  43. [43] Van Der Vaart ( A.) and Wellner ( J.) Weak Convergence and Empirical Processes. Springer, New York (1996). Zbl0862.60002MR1385671

Citations in EuDML Documents

top
  1. Thomas Brodag, Steffen Herbold, Stephan Waack, A Generalized Model of PAC Learning and its Applicability
  2. Vladimir Koltchinskii, Sparsity in penalized empirical risk minimization
  3. Vladimir Koltchinskii, Bounds on margin distributions in learning problems
  4. Peter L. Bartlett, Shahar Mendelson, Petra Philips, On the Optimality of Sample-Based Estimates of the Expectation of the Empirical Minimizer
  5. Antoine Chambaz, Detecting abrupt changes in random fields
  6. Gérard Biau, Estimation de la densité et tests par la méthode combinatoire pénalisée
  7. Antoine Chambaz, Detecting abrupt changes in random fields
  8. Marie Sauve, Christine Tuleau-Malot, Variable selection through CART
  9. Stéphane Boucheron, Olivier Bousquet, Gábor Lugosi, Theory of classification : a survey of some recent advances
  10. Stéphane Boucheron, Olivier Bousquet, Gábor Lugosi, Theory of Classification: a Survey of Some Recent Advances

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.