Wavelet estimation of a density in a GARCH-type model

Christophe Chesneau (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We consider the GARCH-type model: $S=\sigma^2Z$, where $\sigma^2$ and $Z$ are independent random variables. The density of $\sigma^2$ is unknown whereas the one of $Z$ is known. We want to estimate the density of $\sigma^2$ from $n$ observations of $S$ under some dependence assumption (the {\it exponentially strongly mixing dependence}). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence.

Mots clés: GARCH models; Adaptive density estimation; Rate of convergence; Wavelets; Hard thresholding

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

Christophe Chesneau. Wavelet estimation of a density in a GARCH-type model. 2010. ⟨hal-00535837⟩