Wavelet Estimation of a Density in a GARCH-type Model

Abstract

We consider the GARCH-type model: S = σ2 Z, where σ2 and Z are independent random variables. The density of σ2 is unknown whereas the one of Z is known. We want to estimate the density of σ2 from n observations of S under some dependence assumption (the 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.