Combining thresholding rules: a new way to improve the performance of wavelet estimators

Abstract

In this paper, we address the situation where we cannot differentiate wavelet-based threshold procedures because their sets of well-estimated functions (maxisets) are not nested. As a generic solution, we propose to proceed via a combination of these procedures in order to achieve new procedures which perform better in the sense that the involved maxisets contain the union of the previous ones. Throughout the paper we propose illuminating interpretations of the maxiset results and provide conditions to ensure that this combination generates larger maxisets. As an example, we propose to combine vertical- and horizontal-block thresholding procedures that are already known to perform well. We discuss the limitation of our method, and we check our theoretical results through numerical experiments.

Keywords:

• curve estimation
• wavelet methods
• maximal spaces
• rate of convergence
• thresholding rules

AMS Subject Classifications::

• 62G05
• 62G20
• 41A25
• 42C40
• 65T60

Acknowledgements

The authors thank the associate editor and two anonymous referees for the helpful comments and suggestions which led to the considerable improvement of our article. Financial support from the contract ‘Projet d'Actions de Recherche Concertées’ nr. 07/12/002 of the ‘Communauté française de Belgique’ granted by the ‘Académie universitaire Louvain’, the F+11/011 granted by the K.U. Leuven and the FWO G02470.12 are gratefully acknowledged.