Adaptive estimation of the stationary density of discrete and continuous time mixing processes
ESAIM: Probability and Statistics, Tome 6 (2002), pp. 211-238.

In this paper, we study the problem of non parametric estimation of the stationary marginal density f of an α or a β-mixing process, observed either in continuous time or in discrete time. We present an unified framework allowing to deal with many different cases. We consider a collection of finite dimensional linear regular spaces. We estimate f using a projection estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized contrast. We state non asymptotic risk bounds, regarding to the integrated quadratic risk, for our estimators, in both cases of mixing. We show that they are adaptive in the minimax sense over a large class of Besov balls. In discrete time, we also provide a result for model selection among an exponentially large collection of models (non regular case).

DOI : 10.1051/ps:2002012
Classification : 62G07, 62M99
Mots clés : non parametric estimation, projection estimator, adaptive estimation, model selection, mixing processes, continuous time, discrete time 
@article{PS_2002__6__211_0,
     author = {Comte, Fabienne and Merlev\`ede, Florence},
     title = {Adaptive estimation of the stationary density of discrete and continuous time mixing processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {211--238},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2002},
     doi = {10.1051/ps:2002012},
     mrnumber = {1943148},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2002012/}
}
TY  - JOUR
AU  - Comte, Fabienne
AU  - Merlevède, Florence
TI  - Adaptive estimation of the stationary density of discrete and continuous time mixing processes
JO  - ESAIM: Probability and Statistics
PY  - 2002
SP  - 211
EP  - 238
VL  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2002012/
DO  - 10.1051/ps:2002012
LA  - en
ID  - PS_2002__6__211_0
ER  - 
%0 Journal Article
%A Comte, Fabienne
%A Merlevède, Florence
%T Adaptive estimation of the stationary density of discrete and continuous time mixing processes
%J ESAIM: Probability and Statistics
%D 2002
%P 211-238
%V 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2002012/
%R 10.1051/ps:2002012
%G en
%F PS_2002__6__211_0
Comte, Fabienne; Merlevède, Florence. Adaptive estimation of the stationary density of discrete and continuous time mixing processes. ESAIM: Probability and Statistics, Tome 6 (2002), pp. 211-238. doi : 10.1051/ps:2002012. http://www.numdam.org/articles/10.1051/ps:2002012/

[1] G. Banon, Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 (1978) 380-395. | MR | Zbl

[2] G. Banon and H.T. N'Guyen, Recursive estimation in diffusion model. SIAM J. Control Optim. 19 (1981) 676-685. | Zbl

[3] A.R. Barron, L. Birgé and P. Massart, Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-413. | MR | Zbl

[4] H.C.P Berbee, Random walks with stationary increments and renewal theory. Cent. Math. Tracts, Amsterdam (1979). | MR | Zbl

[5] L. Birgé and P. Massart, From model selection to adaptive estimation, in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, edited by D. Pollard, E. Torgersen and G. Yang. Springer-Verlag, New-York (1997) 55-87. | MR | Zbl

[6] L. Birgé and P. Massart, Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 (1998) 329-375. | MR | Zbl

[7] L. Birgé and P. Massart, An adaptive compression algorithm in Besov spaces. Constr. Approx. 16 (2000) 1-36. | MR | Zbl

[8] L. Birgé and Y. Rozenholc, How many bins must be put in a regular histogram? Preprint LPMA 721, http://www.proba.jussieu.fr/mathdoc/preprints/index.html (2002).

[9] D. Bosq, Parametric rates of nonparametric estimators and predictors for continuous time processes. Ann. Stat. 25 (1997) 982-1000. | MR | Zbl

[10] D. Bosq, Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, Second Edition. Springer Verlag, New-York, Lecture Notes in Statist. 110 (1998). | MR | Zbl

[11] D. Bosq and Yu. Davydov, Local time and density estimation in continuous time. Math. Methods Statist. 8 (1999) 22-45. | MR | Zbl

[12] W. Bryc, On the approximation theorem of Berkes and Philipp. Demonstratio Math. 15 (1982) 807-815. | MR | Zbl

[13] C. Butucea, Exact adaptive pointwise estimation on Sobolev classes of densities. ESAIM: P&S 5 (2001) 1-31. | Numdam | MR | Zbl

[14] J.V. Castellana and M.R. Leadbetter, On smoothed probability density estimation for stationary processes. Stochastic Process. Appl. 21 (1986) 179-193. | MR | Zbl

[15] S. Clémençon, Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist. 9 (2000) 323-357. | MR | Zbl

[16] A. Cohen, I. Daubechies and P. Vial, Wavelet and fast wavelet transform on an interval. Appl. Comput. Harmon. Anal. 1 (1993) 54-81. | MR | Zbl

[17] F. Comte and F. Merlevède, Density estimation for a class of continuous time or discretely observed processes. Preprint MAP5 2002-2, http://www.math.infor.univ-paris5.fr/map5/ (2002).

[18] F. Comte and Y. Rozenholc, Adaptive estimation of mean and volatility functions in (auto-)regressive models. Stochastic Process. Appl. 97 (2002) 111-145. | MR | Zbl

[19] I. Daubechies, Ten lectures on wavelets. SIAM: Philadelphia (1992). | MR | Zbl

[20] B. Delyon, Limit theorem for mixing processes, Technical Report IRISA. Rennes (1990) 546.

[21] R.A. Devore and G.G. Lorentz, Constructive approximation. Springer-Verlag (1993). | MR | Zbl

[22] D.L. Donoho and I.M. Johnstone, Minimax estimation with wavelet shrinkage. Ann. Statist. 26 (1998) 879-921. | MR | Zbl

[23] D.L. Donoho, I.M. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding. Ann. Statist. 24 (1996) 508-539. | MR | Zbl

[24] P. Doukhan, Mixing properties and examples. Springer-Verlag, Lecture Notes in Statist. (1995). | MR | Zbl

[25] Y. Efromovich, Nonparametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30 (1985) 557-661. | Zbl

[26] Y. Efromovich and M.S. Pinsker, Learning algorithm for nonparametric filtering. Automat. Remote Control 11 (1984) 1434-1440. | Zbl

[27] G. Kerkyacharian, D. Picard and K. Tribouley, 𝕃 p adaptive density estimation. Bernoulli 2 (1996) 229-247. | MR | Zbl

[28] A.N. Kolmogorov and Y.A. Rozanov, On the strong mixing conditions for stationary Gaussian sequences. Theory Probab. Appl. 5 (1960) 204-207. | Zbl

[29] Y.A. Kutoyants, Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 (1998) 131-155. | Zbl

[30] F. Leblanc, Density estimation for a class of continuous time processes. Math. Methods Statist. 6 (1997) 171-199. | MR | Zbl

[31] H.T. N'Guyen, Density estimation in a continuous-time stationary Markov process. Ann. Statist. 7 (1979) 341-348. | Zbl

[32] E. Rio, The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 (1995) 1188-1203. | MR | Zbl

[33] M. Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42 (1956) 43-47. | MR | Zbl

[34] M. Talagrand, New concentration inequalities in product spaces. Invent. Math. 126 (1996) 505-563. | MR | Zbl

[35] K. Tribouley and G. Viennet, 𝕃 p adaptive density estimation in a β-mixing framework. Ann. Inst. H. Poincaré 34 (1998) 179-208. | Numdam | MR | Zbl

[36] A.Yu. Veretennikov, On hypoellipticity conditions and estimates of the mixing rate for stochastic differential equations. Soviet Math. Dokl. 40 (1990) 94-97. | MR | Zbl

[37] G. Viennet, Inequalities for absolutely regular sequences: Application to density estimation. Probab. Theory Related Fields 107 (1997) 467-492. | MR | Zbl

Cité par Sources :