Inequalities for absolutely regular sequences: Application to density estimation

Probability Theory and Related Fields

Volumn 107, Issue 4, 1997, Pages 467-492

  Viennet, Gabrielle ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031116516     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004400050094     Document Type: Article

References (37)

1. Ango Nze, P., Doukhan, P.: Functional estimation for time series: a general approach. Prépublication 93-43. Université Paris Sud, France 1993
2. Barron, A.R., Birgé, L., Massart, P.: Risk bounds for model selection via penalization. Prépublication 95-54. Université Paris Sud, France 1995
3. Berbee, H.C.P.: Random walks with stationary increments and renewal theory. Cent. Math. Tracts, Amsterdam 1979
4. Birgé, L., Massart, P.: From model selection to adaptative estimation. Prépublication 95-41. Université Paris Sud, France 1995a
5. Birgé, L., Massart, P.: Minimum contrast estimators on sieves. Prépublication 95-42. Université Paris Sud, France 1995b
6. Birgé, L. Massart, P.: An adaptive compression algorithm in Besov spaces. Manuscript 1996
7. Bretagnolle, J., Huber, C.: Estimation des densites: risque minimax. Z.W.V.G. 47, pp. 119-137 (1979)
8. Cenov, N.N.: Evaluation of an unknown distribution density from observations. Soviet Math. 3, pp. 1559-1562 (1962)
9. Delyon, B.: Limit theorem for mixing processes. Tech. Rept. 546. IRISA, Rennes I 1990
10. DeVore, R.A., Lorentz, G.G.: Constructive approximation. Springer, Berlin Heidelberg New York 1993
11. Doob, J.L.: Stochastic processes. Wiley, New York 1953
12. Doukhan, P.: Consistency of delta sequence estimates of a density or of a regression function for a weakly stationnary sequence. Tech. rept. Université Paris Sud, France 1991
13. Doukhan, P.: Mixing: properties and examples. Lecture Notes in Statistics, 72. Springer, Berlin Heidelberg New York 1994
14. Doukhan, P., León, J.R.: Quadratic deviation of projection density estimates. Revista Brasileira Prob. Estatistica, 7, pp. 37-63 (1993)
15. Doukhan, P., Massart, P., Rio, E.: The functional central limit theorem for strongly mixing processes. Ann. Inst. Henri Poincaré, Probab. Stat. 30, 63-82 (1994)
16. Doukhan, P., Massart, P., Rio, E.: Invariance principle for absolutely regular empirical processes. Ann. Inst. Henri Poincaré, Probab. Stat. 31, 393-427 (1995)
17. Földes, A., Revesz, P.: A general method for density estimation. Stud. Sci. Math. Hungar. 9, 81-92 (1974)
18. Ibragimov, I.A.: Some limit theorems for stationary processes. Theor. Probab. Appl. 7, 349-382 (1962)
19. Ibragimov, I.A., Solev, V.N.: A condition for regularity of gaussian stationary processes. Soviet Math. Dokl. 10, 371-375 (1969)
20. Kerkyacharian, G., Picard, D.: Limit of the quadratic risk in density estimation using linear methods. Manuscript, 1995
21. Kolmogorov, A.N., Rozanov, Y.A: On the strong mixing conditions for stationary gaussian sequences. Theor. Probab. Appl. 5, 204-207 (1960)
22. Leblanc, F.: Estimation par ondelettes de la densité marginale d'un processus stochastique: temps dicret, temps continu et discrétisation. Ph.D. thesis, Université de Paris VI 1995
23. Masry, E.: Probability density estimation from dependent observations using wavelets orthonormal bases. Statist. Probab. Lett. 21, 181-194 (1994)
24. Mokkadem, A.: Propriété de mélange des processus autorégressifs polynomiaux. Ann. I.H.P. 26, 133-141 (1990)
25. Parzen, E.J.: On estimating a probability density function and mode. Ann. Math. Statist. 35, 1065-1076 (1962)
26. Pollard, D.: Convergence of stochastic processes. Vol. I. Springer, New York 1984
27. Rio, E.: Covariance inequalities for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. et statist. 29, 587-597 (1993)
28. Rio, E.: The functional law of iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23, 1188-1204 (1995)
29. Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Nat. Ac. Sc. USA 42, 43-47 (1956)
30. p, p > 2 spanned by sequences of independent random variables. Israel J. Math. 8, 273-303 (1970)
31. Roussas, G.G.: Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21, 73-87 (1969)
32. Roussas, G.G.: Nonparametric estimation in mixing sequences of random variables. J. Statist. Plann. Inference 18, 135-149 (1988)
33. Tran, L.T.: Density estimation for time series by histograms. J. Statist. Plann. Inference, 40, 61-79 (1994)
34. Viennet, G.: Estimation Minimax et adaptative dans un cadre absolument régulier. Ph.D. thesis, Université Paris XI, Orsay 1995
35. Vieu, P.: Quadratic errors for nonparameteric estimates under dependence. J. Multivariate. Anal. 39, 324-347 (1991)
36. Walter, G., Blum, J.: Probability density estimation using Delta sequences. Ann. Statist. 42, 328-340 (1979)
37. Yu, B.: Density estimation in the uniform-norm for dependant data with applications to the Gibbs Sampler. Ann. Statist. 21, 711-735 (1993)