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Adaptive estimation of the stationary densityof discrete and continuous time mixingprocesses

Published online by Cambridge University Press:  15 November 2002

Fabienne Comte  and
Florence Merlevède
Fabienne Comte
Affiliation:
Université Paris V, Laboratoire MAP5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; [email protected].
Florence Merlevède
Affiliation:
Université Paris VI, LSTA, 4 place Jussieu, 75252 Paris Cedex 05, France; [email protected].
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Abstract

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).

Keywords

Non parametric estimationprojection estimatoradaptive estimationmodel selectionmixing processescontinuoustimediscrete time.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 211 - 238
Copyright
© EDP Sciences, SMAI, 2002

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References

Banon, G., Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 (1978) 380-395. CrossRef
Banon, G. and N'Guyen, H.T., Recursive estimation in diffusion model. SIAM J. Control Optim. 19 (1981) 676-685. CrossRef
Barron, A.R., Birgé, L. and Massart, P., Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-413. CrossRef
H.C.P Berbee, Random walks with stationary increments and renewal theory. Cent. Math. Tracts, Amsterdam (1979).
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.
Birgé, L. and Massart, P., Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 (1998) 329-375. CrossRef
Birgé, L. and Massart, P., An adaptive compression algorithm in Besov spaces. Constr. Approx. 16 (2000) 1-36. CrossRef
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).
Bosq, D., Parametric rates of nonparametric estimators and predictors for continuous time processes. Ann. Stat. 25 (1997) 982-1000. CrossRef
D. Bosq, Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, Second Edition. Springer Verlag, New-York, Lecture Notes in Statist. 110 (1998).
Bosq, D. and Davydov, Yu., Local time and density estimation in continuous time. Math. Methods Statist. 8 (1999) 22-45.
Bryc, W., On the approximation theorem of Berkes and Philipp. Demonstratio Math. 15 (1982) 807-815.
Butucea, C., Exact adaptive pointwise estimation on Sobolev classes of densities. ESAIM: P&S 5 (2001) 1-31. CrossRef
Castellana, J.V. and Leadbetter, M.R., On smoothed probability density estimation for stationary processes. Stochastic Process. Appl. 21 (1986) 179-193. CrossRef
Clémençon, S., Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist. 9 (2000) 323-357.
Cohen, A., Daubechies, I. and Vial, P., Wavelet and fast wavelet transform on an interval. Appl. Comput. Harmon. Anal. 1 (1993) 54-81. CrossRef
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).
Comte, F. and Rozenholc, Y., Adaptive estimation of mean and volatility functions in (auto-)regressive models. Stochastic Process. Appl. 97 (2002) 111-145. CrossRef
I. Daubechies, Ten lectures on wavelets. SIAM: Philadelphia (1992).
B. Delyon, Limit theorem for mixing processes, Technical Report IRISA. Rennes (1990) 546.
R.A. DeVore and G.G. Lorentz, Constructive approximation. Springer-Verlag (1993).
Donoho, D.L. and Johnstone, I.M., Minimax estimation with wavelet shrinkage. Ann. Statist. 26 (1998) 879-921.
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D., Density estimation by wavelet thresholding. Ann. Statist. 24 (1996) 508-539.
P. Doukhan, Mixing properties and examples. Springer-Verlag, Lecture Notes in Statist. (1995).
Efromovich, Y., Nonparametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30 (1985) 557-661. CrossRef
Efromovich, Y. and Pinsker, M.S., Learning algorithm for nonparametric filtering. Automat. Remote Control 11 (1984) 1434-1440.
Kerkyacharian, G., Picard, D. and Tribouley, K., $\mathbb{L}_p$adaptive density estimation. Bernoulli 2 (1996) 229-247.
Kolmogorov, A.N. and Rozanov, Y.A., On the strong mixing conditions for stationary Gaussian sequences. Theory Probab. Appl. 5 (1960) 204-207. CrossRef
Kutoyants, Y.A., Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 (1998) 131-155. CrossRef
Leblanc, F., Density estimation for a class of continuous time processes. Math. Methods Statist. 6 (1997) 171-199.
N'Guyen, H.T., Density estimation in a continuous-time stationary Markov process. Ann. Statist. 7 (1979) 341-348. CrossRef
Rio, E., The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 (1995) 1188-1203. CrossRef
Rosenblatt, M., A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42 (1956) 43-47. CrossRef
Talagrand, M., New concentration inequalities in product spaces. Invent. Math. 126 (1996) 505-563. CrossRef
Tribouley, K. and Viennet, G., $\mathbb{L}_p$ adaptive density estimation in a $\beta$-mixing framework. Ann. Inst. H. Poincaré 34 (1998) 179-208. CrossRef
Veretennikov, A.Yu., On hypoellipticity conditions and estimates of the mixing rate for stochastic differential equations. Soviet Math. Dokl. 40 (1990) 94-97.
Viennet, G., Inequalities for absolutely regular sequences: Application to density estimation. Probab. Theory Related Fields 107 (1997) 467-492. CrossRef