Adaptive density estimation under weak dependence

¹ Laboratoire Jean Kuntzmann, INP Grenoble, 38041 Grenoble Cedex 9, France
² SAMOS-MATISSE (Statistique Appliquée et Modélisation Stochastique), Centre d'Économie de la Sorbonne Université Paris 1 – Panthéon-Sorbonne, CNRS 90, Rue de Tolbiac, 75634 Paris Cedex 13, France

Corresponding authors: irene.gannaz@imag.fr olivier.wintenberger@univ-paris1.fr

Received: 9 October 2006
Revised: 16 January 2008
Revised: 11 July 2008

Abstract

Assume that (X_t)_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 (X_t)_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.