Adaptative density estimation with dependent observations
Résumé
Assume that (X_n) is a real valued stationary time series admitting a common density f. To estimate f in an independent and identically distributed setting, Donoho, Johnstone, Kerkyacharian & Picard (1996) proposed a quasi-minimax method based on thresholding wavelets. The aim of the present work is to extend this methodology to the dependent case. For this purpose, we introduce the new Phi-weak dependence based on a probability inequality, which includes a large spectrum of classical weak dependence cases. Actually, we establish a link between this condition and the $\tilde \phi$-dependence of Dedecker & Prieur (2004) and the $\eta$-weak dependence condition introduced by Doukhan & Louhichi (1999). The estimator we propose adapts the threshold to the dependence of the observations. We obtain near minimax convergence rates for L^p losses, p>= 1. We thus apply this method on simulations of non stationary but geometrically ergodic cases like dynamical systems and Markovian fields on the line.