On wavelet analysis and some distribution results

This thesis is concerned with distribution theory as well as hypothesis testing and inference for wavelet shrinkage methods in nonparametric regression. Attention is restricted to wavelet VisuShrink and SureShrink with equi-spaced design points. The distribution of L2-risk for the wavelet VisuShrink estimator is shown to be asymptotically normal, under simple regularity conditions. Furthermore, the distribution of L2-risk for the wavelet SureShrink estimator based on theoretical thresholding is shown to be asymptotically normal. For any L2 function, the parallel result for the wavelet SureShrink estimator based on empirical thresholding is also obtained. Nonparametric testing and inference have been the focus of much interest recently. The derived distribution theory for the wavelet estimators, together with the Bootstrap principle, is used to drive a nonparametric testing procedure. Testing within this framework makes the distribution results useful to making confidence statements concerning an unknown function (object) in signal processing. Simulations are performed to illustrate the distribution theory and the testing procedure. In addition, the normal approximation for finite sample sizes is examined, which suggests that the normal approximation is reasonably good even for small sample sizes