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 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 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.
Mots-clés : 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}, publisher = {EDP-Sciences}, volume = {14}, year = {2010}, doi = {10.1051/ps:2008025}, mrnumber = {2654551}, language = {en}, url = {http://archive.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 - http://archive.numdam.org/articles/10.1051/ps:2008025/ DO - 10.1051/ps:2008025 LA - en ID - PS_2010__14__151_0 ER -
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. http://archive.numdam.org/articles/10.1051/ps:2008025/
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