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Block-threshold-adapted Estimators via a Maxiset Approach

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  • Florent Autin
  • Jean-Marc Freyermuth
  • Rainer Von Sachs

Abstract

type="main" xml:id="sjos12012-abs-0001"> We study the maxiset performance of a large collection of block thresholding wavelet estimators, namely the horizontal block thresholding family. We provide sufficient conditions on the choices of rates and threshold values to ensure that the involved adaptive estimators obtain large maxisets. Moreover, we prove that any estimator of such a family reconstructs the Besov balls with a near-minimax optimal rate that can be faster than the one of any separable thresholding estimator. Then, we identify, in particular cases, the best estimator of such a family, that is, the one associated with the largest maxiset. As a particularity of this paper, we propose a refined approach that models method-dependent threshold values. By a series of simulation studies, we confirm the good performance of the best estimator by comparing it with the other members of its family.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:41:y:2014:i:1:p:240-258
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    File URL: http://hdl.handle.net/10.1111/sjos.12012
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Antoniadis, Anestis & Bigot, Jeremie & Sapatinas, Theofanis, 2001. "Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 6(i06).
    4. Autin, Florent & Freyermuth, Jean-Marc & von Sachs, Rainer, 2012. "Combining thresholding rules: a new way to improve the performance of wavelet estimators," LIDAM Reprints ISBA 2012025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Chesneau, Christophe, 2008. "On the maxiset comparison between hard and block thresholding methods," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 675-681, April.
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