Nonparametric estimation of the derivatives of the stationary density for stationary processes
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 33-69.

In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete times t = 0,Δ,...,nΔ. The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f(j) belongs to the Besov space 2 , α , then our estimator converges at rate ()-α/(2α+2j+1). Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f′. When the sampling interval Δ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is ()-α/(2α+1). When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.

DOI : 10.1051/ps/2011102
Classification : 62G05, 60G10
Mots clés : derivatives of the stationary density, diffusion processes, mixing processes, nonparametric estimation, stationary processes
@article{PS_2013__17__33_0,
     author = {Schmisser, Emeline},
     title = {Nonparametric estimation of the derivatives of the stationary density for stationary processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {33--69},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011102},
     mrnumber = {3002995},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2011102/}
}
TY  - JOUR
AU  - Schmisser, Emeline
TI  - Nonparametric estimation of the derivatives of the stationary density for stationary processes
JO  - ESAIM: Probability and Statistics
PY  - 2013
SP  - 33
EP  - 69
VL  - 17
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2011102/
DO  - 10.1051/ps/2011102
LA  - en
ID  - PS_2013__17__33_0
ER  - 
%0 Journal Article
%A Schmisser, Emeline
%T Nonparametric estimation of the derivatives of the stationary density for stationary processes
%J ESAIM: Probability and Statistics
%D 2013
%P 33-69
%V 17
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2011102/
%R 10.1051/ps/2011102
%G en
%F PS_2013__17__33_0
Schmisser, Emeline. Nonparametric estimation of the derivatives of the stationary density for stationary processes. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 33-69. doi : 10.1051/ps/2011102. http://www.numdam.org/articles/10.1051/ps/2011102/

[1] S. Arlot and P. Massart, Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245-279.

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

[3] A. Beskos, O. Papaspiliopoulos and G.O. Roberts, Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 (2006) 1077-1098. | MR | Zbl

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

[5] F. Comte and F. Merlevède, Adaptive estimation of the stationary density of discrete and continuous time mixing processes. ESAIM : PS 6 (2002) 211-238 (electronic). New directions in time series analysis. Luminy (2001). | Numdam | MR

[6] F. Comte and F. Merlevède, Super optimal rates for nonparametric density estimation via projection estimators. Stoc. Proc. Appl. 115 (2005) 797-826. | MR | Zbl

[7] F. Comte, Y. Rozenholc and M.L. Taupin, Penalized contrast estimator for adaptive density deconvolution. Can. J. Stat. 34 (2006) 431-452. | MR | Zbl

[8] F. Comte, V. Genon-Catalot and Y. Rozenholc, Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514-543. | MR | Zbl

[9] A.S. Dalalyan and Y.A. Kutoyants, Asymptotically efficient estimation of the derivative of the invariant density. Stat. Inference Stoch. Process. 6 (2003) 89-107. | MR | Zbl

[10] R.A. Devore and G.G. Lorentz, Constructive approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 303 (1993). | MR | Zbl

[11] A. Gloter, Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM : PS 4 (2000) 205-227. | EuDML | Numdam | MR | Zbl

[12] E. Gobet, M. Hoffmann and M. Reiß, Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Stat. 32 (2004) 2223-2253. | MR | Zbl

[13] N. Hosseinioun, H. Doosti and H.A. Niroumand, Wavelet-based estimators of the integrated squared density derivatives for mixing sequences. Pakistan J. Stat. 25 (2009) 341-350. | MR

[14] C. Lacour, Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées. Ph.D. thesis, Université Paris Descartes (2007).

[15] C. Lacour, Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stoc. Proc. Appl. 118 (2008) 232-260. | MR | Zbl

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

[17] M. Lerasle, Adaptive density estimation of stationary β-mixing and τ-mixing processes. Math. Methods Stat. 18 (2009) 59-83. | MR | Zbl

[18] M. Lerasle, Optimal model selection for stationary data under various mixing conditions (2010). | Zbl

[19] E. Masry, Probability density estimation from dependent observations using wavelets orthonormal bases. Stat. Probab. Lett. 21 (1994) 181-194. Available on : http://dx.doi.org/10.1016/0167-7152(94)90114-7. | MR | Zbl

[20] Y. Meyer, Ondelettes et opérateurs I. Actualités Mathématiques [Current Mathematical Topics]. Hermann, Paris, Ondelettes [Wavelets] (1990). | Zbl

[21] E. Pardoux and A.Y. Veretennikov, On the Poisson equation and diffusion approximation I. Ann. Probab. 29 (2001) 1061-1085. | MR | Zbl

[22] B.L.S.P. Rao, Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernet. 28 (1996) 91-100. | MR | Zbl

[23] E. Schmisser, Non-parametric drift estimation for diffusions from noisy data. Stat. Decis. 28 (2011) 119-150. | MR | Zbl

[24] G. Viennet, Inequalities for absolutely regular sequences : application to density estimation. Probab. Theory Relat. Fields (1997). | MR | Zbl

Cité par Sources :