Data adaptive wavelet methods for Gaussian long-memory processes

Abstract

In this thesis, we investigate some adaptive wavelet approaches for a so-called nonparametric regression model with strongly dependent Gaussian residuals. At first, we discuss data adaptive wavelet estimation of a trend function. It turns out that under certain smoothing conditions on the trend function, the asymptotic rate of the mean integrated square error (MISE) of a trend estimator obtained by a hard wavelet thresholding is the same as for a linear wavelet estimator. Asymptotic expressions for the optimalMISE and corresponding optimal smoothing and resolution parameters are derived. Furthermore, we focus on the non-continuous trend functions and derive corresponding optimal smoothing, resolution and thresholding parameters. Due to adaptation to the properties of the underlying trend function, the approach shows very good performance for piecewise smooth trend functions while remaining competitive with minimax wavelet estimation for functions with discontinuities. It turns out that the same expression for MISE still holds and the hard thresholding wavelet estimator can be understood as a combination of two components, a smoothing component consisting of a certain number of lower resolution levels where no thresholding is applied, and a higher resolution component filtered by thresholding procedure. The first component leads to good performance for smooth functions, whereas the second component is useful for modeling discontinuities. This fact is used to develop an appropriate test for the null hypothesis that the trend is continuous against the alternative that it has at least one isolated jump. The proposed test statistic is based on blockwise resampling of estimated residual variances. Asymptotic validity of the test is derived. Simulations illustrate the asymptotic results and finite sample behavior of the proposed methods.