Adaptive density estimation under weak dependence
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 151-172

Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat. 24 (1996) 508-539] propose near-minimax estimators f ^ n based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators f ^ n depend on weak dependence properties of the sequence (Xt)t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

DOI : 10.1051/ps:2008025
Classification : 62G07, 60G10, 60G99, 62G20
Keywords: adaptive estimation, cross validation, hard thresholding, near minimax results, nonparametric density estimation, soft thresholding, wavelets, weak dependence 
@article{PS_2010__14__151_0,
     author = {Gannaz, Ir\`ene and Wintenberger, Olivier},
     title = {Adaptive density estimation under weak dependence},
     journal = {ESAIM: Probability and Statistics},
     pages = {151--172},
     year = {2010},
     publisher = {EDP Sciences},
     volume = {14},
     doi = {10.1051/ps:2008025},
     mrnumber = {2654551},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2008025/}
}
TY  - JOUR
AU  - Gannaz, Irène
AU  - Wintenberger, Olivier
TI  - Adaptive density estimation under weak dependence
JO  - ESAIM: Probability and Statistics
PY  - 2010
SP  - 151
EP  - 172
VL  - 14
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps:2008025/
DO  - 10.1051/ps:2008025
LA  - en
ID  - PS_2010__14__151_0
ER  - 
%0 Journal Article
%A Gannaz, Irène
%A Wintenberger, Olivier
%T Adaptive density estimation under weak dependence
%J ESAIM: Probability and Statistics
%D 2010
%P 151-172
%V 14
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps:2008025/
%R 10.1051/ps:2008025
%G en
%F PS_2010__14__151_0
Gannaz, Irène; Wintenberger, Olivier. Adaptive density estimation under weak dependence. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 151-172. doi: 10.1051/ps:2008025

[1] D. Andrews, Non strong mixing autoregressive processes. J. Appl. Probab. 21 (1984) 930-934. | Zbl

[2] D. Bosq and D. Guegan, Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system. Stat. Probab. Lett. 25 (1995) 201-212. | Zbl

[3] F. Comte and F. Merlevède, Adaptive estimation of the stationary density of discrete and continuous time mixing processes. ESAIM: PS 6 (2002) 211-238.

[4] I. Daubechies, Ten Lectures on Wavelets, volume 61. SIAM Press (1992). | Zbl

[5] J. Dedecker and C. Prieur, New dependence coefficients: Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203-235. | Zbl

[6] J. Dedecker and C. Prieur, An empirical central limit theorem for dependent sequences. Stoch. Process. Appl. 117 (2007) 121-142. | Zbl

[7] J. Dedecker, P. Doukhan, G. Lang, J.R. Leon, S. Louhichi and C. Prieur, Weak Dependence: Models, Theory and Applications. Springer-Verlag (2007). | Zbl

[8] D. Donoho, I. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding. Ann. Stat. 24 (1996) 508-539. | Zbl

[9] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84 (1999) 313-342. | Zbl

[10] P. Doukhan and M. Neumann, A Bernstein type inequality for times series. Stoch. Process. Appl. 117 (2007) 878-903. | Zbl

[11] P. Doukhan, G. Teyssière and P. Winant, Vector valued ARCH infinity processes, in Dependence in Probability and Statistics . Lect. Notes Statist. Springer, New York (2006). | Zbl

[12] P. Doukhan and L. Truquet, A fixed point approach to model random fields. Alea 2 (2007) 111-132. | Zbl

[13] P. Doukhan and O. Wintenberger, Weakly dependent chains with infinite memory. Stoch. Process. Appl. 118 (2008) 1997-2013. | Zbl

[14] P. Doukhan and O. Wintenberger, Invariance principle for new weakly dependent stationary models. Probab. Math. Statist. 27 (2007) 45-73. | Zbl

[15] S. Gouëzel, Central limit theorem and stable laws for intermittent maps. Probab. Theory Relat. Fields 128 (2004) 82-122. | Zbl

[16] W. Hardle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelets Approximation and Statistical Applications. Lect. Notes Statist. 129. Springer-Verlag (1998). | Zbl

[17] A. Juditsky and S. Lambert-Lacroix, On minimax density estimation on ℝ. Bernoulli, 10 (2004) 187-220. | Zbl

[18] C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency. Ergodic Theory Dynam. Syst. 19 (1999) 671-686. | Zbl

[19] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Machine Intelligence 11 (1989) 674-693. | Zbl

[20] V. Maume-Deschamps, Exponential inequalities and functional estimations for weak dependent data; applications to dynamical systems. Stoch. Dynam. 6 (2006) 535-560. | Zbl

[21] Y. Meyer, Wavelets and Operators. Cambridge University Press (1992). | Zbl

[22] C. Prieur, Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques. Ph.D. thesis, CREST, 2001.

[23] N. Ragache and O. Wintenberger, Convergence rates for density estimators of weakly dependent time series, in Dependence in Probability and Statistics , P. Bertail, P. Doukhan, and P. Soulier (Eds.). Lect. Notes Statist. 187. Springer, New York (2006), pp. 349-372. | Zbl

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

[25] M.-L. Vanharen, Estimation par ondelettes dans les systèmes dynamiques. C. R. Acad. Sci. Paris 342 (2006) 523-525. | Zbl

[26] M. Vannucci, Nonparametric density estimation using wavelets. Tech. Rep., Texas A and M University, 1998.

[27] M. Viana, Stochastic dynamics of deterministic systems. Available at http://w3.impa.br/~viana (1997).

[28] B. Vidakovic, Pollen bases and Daubechies-Lagarias algorithm in MATLAB (2002). Available at http://www2.isye.gatech.edu/~brani/datasoft/DL.pdf.

[29] Wavelab. http://www-stat.stanford.edu/~wavelab/.

[30] L. Young, Recurrence times and rates of mixing. Isr. J. Math. 110 (1999) 0021-2172. | Zbl

Cité par Sources :