Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets (Journal of Computational Mathematics, 2021)
This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.
Keywords: Wavelets; Point-wise risk; Thresholding; Data-driven; GARCH-type model.