Tree-structured wavelets in nonparametric function estimation

Wavelet thresholding methods, especially those which pool information from geometric structures in the coefficient domain, are known to be powerful for nonparametric function estimation. In this thesis, we focus on a family of Tree-Structured Wavelet (TSW) estimators so called Vertical Block Thresholding (VBT) family. For each estimator we provide the maximal functional space (maxiset) for which the quadratic risk reaches a given rate of convergence. We identify the ideal estimator of this family, that is the one associated with the largest maxiset and we emphasize the importance of considering method-dependent threshold values. While it is a current research topic for the VBT family, we address this problem in the similar but simpler context of the nonoverlapping Horizontal Block Thresholding family. We next study the situation where we cannot differentiate wavelet-based estimators because their maxisets are not nested. As a generic solution, we propose to proceed via a combination of these estimators in order to achieve new estimators which perform better in the sense that the involved maxisets contain the union of the previous ones. Finally, we use the relation between TSW and recursive dyadic partitioning to develop a novel method for estimating the spectrum of a stationary process using time series traces recorded from experimental designs. Our procedure estimates the “common” log-spectrum and the variability over the traces (or subjects) using a mixed effects model. Numerical studies and a real data example confirm that the proposed methods perform very well.(STAT 3) -- UCL, 201