ADAPTIVE DENSITY ESTIMATION FOR GENERAL ARCH MODELS

We consider a model Yt stht in which ~st ! is not independent of the noise process ~ht ! but st is independent of ht for each t+ We assume that ~st ! is stationary, and we propose an adaptive estimator of the density of ln~st 2! based on the observations Yt + Under a new dependence structure, the t-dependency defined by Dedecker and Prieur ~2005, Probability Theory and Related Fields 132, 203–236!, we prove that the rates of this nonparametric estimator coincide with the rates obtained in the independent and identically distributed ~i+i+d+! case when ~st ! and ~ht ! are independent+ The results apply to various linear and nonlinear general autoregressive conditionally heteroskedastic ~ARCH! processes+ They are illustrated by simulations applying the deconvolution algorithm of Comte, Rozenholc, and Taupin ~2006, Canadian Journal of Statistics 34, 431– 452! to a new noise density+