Adaptive density estimation for general ARCH models

Fabienne Comte (1) , Jérôme Dedecker (2) , Marie-Luce Taupin (1, 3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 LSTA - Laboratoire de Statistique Théorique et Appliquée
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider a model $Y_t=\sigma_t\eta_t$ in which $(\sigma_t)$ is not independent of the noise process $(\eta_t)$, but $\sigma_t$ is independent of $\eta_t$ for each $t$. We assume that $(\sigma_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(\sigma^2_t)$ based on the observations $Y_t$. Under various dependence structures, the rates of this nonparametric estimator coincide with the minimax rates obtained in the i.i.d. case when $(\sigma_t)$ and $(\eta_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes.

Mots clés: Adaptive density estimation; Penalized contrast.

Mots clés: Model selection; Penalized contrast; Deconvolution; General ARCH models

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Fabienne Comte, Jérôme Dedecker, Marie-Luce Taupin. Adaptive density estimation for general ARCH models. 2006. ⟨hal-00101417⟩