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 , then our estimator converges at rate (nΔ)-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 (nΔ)-α/(2α+1). When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.
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] Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245-279.
and ,[2] Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999) 301-413. | MR | Zbl
, and ,[3] Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 (2006) 1077-1098. | MR | Zbl
, and ,[4] Parametric rates of nonparametric estimators and predictors for continuous time processes. Ann. Stat. 25 (1997) 982-1000. | MR | Zbl
,[5] 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
and ,[6] Super optimal rates for nonparametric density estimation via projection estimators. Stoc. Proc. Appl. 115 (2005) 797-826. | MR | Zbl
and ,[7] Penalized contrast estimator for adaptive density deconvolution. Can. J. Stat. 34 (2006) 431-452. | MR | Zbl
, and ,[8] Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514-543. | MR | Zbl
, and ,[9] Asymptotically efficient estimation of the derivative of the invariant density. Stat. Inference Stoch. Process. 6 (2003) 89-107. | MR | Zbl
and ,[10] Constructive approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 303 (1993). | MR | Zbl
and ,[11] 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] Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Stat. 32 (2004) 2223-2253. | MR | Zbl
, and ,[13] Wavelet-based estimators of the integrated squared density derivatives for mixing sequences. Pakistan J. Stat. 25 (2009) 341-350. | MR
, and ,[14] Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées. Ph.D. thesis, Université Paris Descartes (2007).
,[15] Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stoc. Proc. Appl. 118 (2008) 232-260. | MR | Zbl
,[16] Density estimation for a class of continuous time processes. Math. Methods Stat. 6 (1997) 171-199. | MR | Zbl
,[17] Adaptive density estimation of stationary β-mixing and τ-mixing processes. Math. Methods Stat. 18 (2009) 59-83. | MR | Zbl
,[18] Optimal model selection for stationary data under various mixing conditions (2010). | Zbl
,[19] 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] Ondelettes et opérateurs I. Actualités Mathématiques [Current Mathematical Topics]. Hermann, Paris, Ondelettes [Wavelets] (1990). | Zbl
,[21] On the Poisson equation and diffusion approximation I. Ann. Probab. 29 (2001) 1061-1085. | MR | Zbl
and ,[22] Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernet. 28 (1996) 91-100. | MR | Zbl
,[23] Non-parametric drift estimation for diffusions from noisy data. Stat. Decis. 28 (2011) 119-150. | MR | Zbl
,[24] Inequalities for absolutely regular sequences : application to density estimation. Probab. Theory Relat. Fields (1997). | MR | Zbl
,Cité par Sources :