Adaptive density estimation under dependence

Abstract

Assume that $(X_t{)}_{t\in \mathbb{Z}}$ is a real valued time series admitting a common marginal density $f$ with respect to Lebesgue's measure. Donoho {\it et al.} (1996) propose a near-minimax method 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 ${\hat{f}}_n$ depend on weak dependence properties of the sequence $(X_t{)}_{t\in \mathbb{Z}}$ 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.

Cited by in corpus (5)

• Adaptive density estimation for stationary processes
• Optimal model selection for density estimation of stationary data under various mixing conditions
• Adaptive non-parametric estimation in the presence of dependence
• Adaptive directional estimator of the density in R^d for independent and mixing sequences
• On the integrated mean squared error of wavelet density estimation for linear processes