Adaptive density estimation under weak dependence

Irène Gannaz; Olivier Wintenberger

ESAIM: Probability and Statistics (2010)

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

Abstract

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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.

How to cite

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Gannaz, 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 -

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