# Adaptive density estimation under weak dependence

Irène Gannaz; Olivier Wintenberger

ESAIM: Probability and Statistics (2010)

- Volume: 14, page 151-172
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topGannaz, Irène, and Wintenberger, Olivier. "Adaptive density estimation under weak dependence." ESAIM: Probability and Statistics 14 (2010): 151-172. <http://eudml.org/doc/250826>.

@article{Gannaz2010,

abstract = {
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 $\widehat f_n$ 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 $\widehat f_n$ 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.
},

author = {Gannaz, Irène, Wintenberger, Olivier},

journal = {ESAIM: Probability and Statistics},

keywords = {Adaptive estimation; cross validation; hard thresholding; near minimax results; nonparametric density estimation; soft thresholding; wavelets; weak dependence; adaptive estimation},

language = {eng},

month = {5},

pages = {151-172},

publisher = {EDP Sciences},

title = {Adaptive density estimation under weak dependence},

url = {http://eudml.org/doc/250826},

volume = {14},

year = {2010},

}

TY - JOUR

AU - Gannaz, Irène

AU - Wintenberger, Olivier

TI - Adaptive density estimation under weak dependence

JO - ESAIM: Probability and Statistics

DA - 2010/5//

PB - EDP Sciences

VL - 14

SP - 151

EP - 172

AB -
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 $\widehat f_n$ 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 $\widehat f_n$ 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.

LA - eng

KW - Adaptive estimation; cross validation; hard thresholding; near minimax results; nonparametric density estimation; soft thresholding; wavelets; weak dependence; adaptive estimation

UR - http://eudml.org/doc/250826

ER -

## References

top- D. Andrews, Non strong mixing autoregressive processes. J. Appl. Probab.21 (1984) 930–934. Zbl0552.60049
- 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. Zbl0841.62028
- F. Comte and F. Merlevède, Adaptive estimation of the stationary density of discrete and continuous time mixing processes. ESAIM: PS6 (2002) 211–238.
- I. Daubechies, Ten Lectures on Wavelets, volume 61. SIAM Press (1992). Zbl0776.42018
- J. Dedecker and C. Prieur, New dependence coefficients: Examples and applications to statistics. Probab. Theory Relat. Fields132 (2005) 203–235. Zbl1061.62058
- J. Dedecker and C. Prieur, An empirical central limit theorem for dependent sequences. Stoch. Process. Appl.117 (2007) 121–142. Zbl1117.60035
- J. Dedecker, P. Doukhan, G. Lang, J.R. Leon, S. Louhichi and C. Prieur, Weak Dependence: Models, Theory and Applications. Springer-Verlag (2007). Zbl1165.62001
- D. Donoho, I. Johnstone, G. Kerkyacharian and D. Picard, Density estimation by wavelet thresholding. Ann. Stat.24 (1996) 508–539. Zbl0860.62032
- P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl.84 (1999) 313–342. Zbl0996.60020
- P. Doukhan and M. Neumann, A Bernstein type inequality for times series. Stoch. Process. Appl.117 (2007) 878–903. Zbl1117.60018
- 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).
- P. Doukhan and L. Truquet, A fixed point approach to model random fields. Alea2 (2007) 111–132. Zbl1140.60029
- P. Doukhan and O. Wintenberger, Weakly dependent chains with infinite memory. Stoch. Process. Appl.118 (2008) 1997–2013. Zbl1166.60031
- P. Doukhan and O. Wintenberger, Invariance principle for new weakly dependent stationary models. Probab. Math. Statist.27 (2007) 45–73. Zbl1124.60031
- S. Gouëzel, Central limit theorem and stable laws for intermittent maps. Probab. Theory Relat. Fields128 (2004) 82–122. Zbl1038.37007
- W. Hardle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelets Approximation and Statistical Applications. Lect. Notes Statist. 129. Springer-Verlag (1998). Zbl0899.62002
- A. Juditsky and S. Lambert-Lacroix, On minimax density estimation on $\mathbb{R}$. Bernoulli, 10 (2004) 187–220. Zbl1076.62037
- C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency. Ergodic Theory Dynam. Syst.19 (1999) 671–686. Zbl0988.37035
- S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Machine Intelligence11 (1989) 674–693. Zbl0709.94650
- V. Maume-Deschamps, Exponential inequalities and functional estimations for weak dependent data; applications to dynamical systems. Stoch. Dynam.6 (2006) 535–560. Zbl1130.37319
- Y. Meyer, Wavelets and Operators. Cambridge University Press (1992). Zbl0776.42019
- C. Prieur, Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques. Ph.D. thesis, CREST, 2001.
- 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. Zbl1113.62055
- K. Tribouley and G. Viennet, ${\text{L}}_{p}$-adaptive density estimation in a β-mixing framework. Ann. Inst. H. Poincaré, B34 (1998) 179–208. Zbl0941.62041
- M.-L. Vanharen, Estimation par ondelettes dans les systèmes dynamiques. C. R. Acad. Sci. Paris342 (2006) 523–525. Zbl1086.62089
- M. Vannucci, Nonparametric density estimation using wavelets. Tech. Rep., Texas A and M University, 1998.
- M. Viana, Stochastic dynamics of deterministic systems. Available at (1997). URIhttp://w3.impa.br/~viana
- B. Vidakovic, Pollen bases and Daubechies-Lagarias algorithm in MATLAB (2002). Available at . URIhttp://www2.isye.gatech.edu/~brani/datasoft/DL.pdf
- Wavelab. . URIhttp://www-stat.stanford.edu/~wavelab/
- L. Young, Recurrence times and rates of mixing. Isr. J. Math.110 (1999) 0021–2172.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.