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

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  • F. Autin
  • J.-M. Freyermuth
  • R. von Sachs

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.

Suggested Citation

  • F. Autin & J.-M. Freyermuth & R. von Sachs, 2012. "Combining thresholding rules: a new way to improve the performance of wavelet estimators," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 905-922, December.
    • Handle: RePEc:taf:gnstxx:v:24:y:2012:i:4:p:905-922
    DOI: 10.1080/10485252.2012.709854
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    References listed on IDEAS

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    1. Gérard Kerkyacharian & Dominique Picard & Lucien Birgé & Peter Hall & Oleg Lepski & Enno Mammen & Alexandre Tsybakov & G. Kerkyacharian & D. Picard, 2000. "Thresholding algorithms, maxisets and well-concentrated bases," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 283-344, December.
    2. Autin, F. & Freyermuth, Jean-Marc & von Sachs, Rainer, 2011. "Ideal denoising within a family of tree-structured wavelet estimators," LIDAM Discussion Papers ISBA 2011002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Engel, J., 1994. "A Simple Wavelet Approach to Nonparametric Regression from Recursive Partitioning Schemes," Journal of Multivariate Analysis, Elsevier, vol. 49(2), pages 242-254, May.
    4. Cai, T. Tony, 2008. "On information pooling, adaptability and superefficiency in nonparametric function estimation," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 421-436, March.
    5. Autin, Florent & Freyermuth, Jean-Marc & von Sachs, Rainer, 2011. "Ideal denoising within a family of tree-structured wavelet estimators," LIDAM Reprints ISBA 2011037, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Stuart Barber & Guy P. Nason, 2004. "Real nonparametric regression using complex wavelets," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 927-939, November.
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    1. Florent Autin & Jean-Marc Freyermuth & Rainer Von Sachs, 2014. "Block-threshold-adapted Estimators via a Maxiset Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 240-258, March.

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