ADAPTIVE DENSITY ESTIMATION FOR GENERAL ARCH MODELS
We consider a model <italic>Y</italic><sub>null</sub> = σ<sub>null</sub>η<sub>null</sub> in which (σ<sub>null</sub>) is not independent of the noise process (η<sub>null</sub>) but σ<sub>null</sub> is independent of η<sub>null</sub> for each <italic>t</italic>. We assume that (σ<sub>null</sub>) is stationary, and we propose an adaptive estimator of the density of ln(σ<sub>null</sub><sup>2</sup>) based on the observations <italic>Y</italic><sub>null</sub>. Under a new dependence structure, the τ-dependency defined by Dedecker and Prieur (2005, <italic>Probability Theory and Related Fields</italic> 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 (σ<sub>null</sub>) and (η<sub>null</sub>) 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, <italic>Canadian Journal of Statistics</italic> 34, 431–452) to a new noise density.
