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The root–unroot algorithm for density estimation as implemented via wavelet block thresholding
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  • Published: 06 January 2009

The root–unroot algorithm for density estimation as implemented via wavelet block thresholding

  • Lawrence Brown1,
  • Tony Cai1,
  • Ren Zhang1,
  • Linda Zhao1 &
  • …
  • Harrison Zhou2 

Probability Theory and Related Fields volume 146, pages 401–433 (2010)Cite this article

  • 341 Accesses

  • 44 Citations

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Abstract

We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression estimators could then be applied to the transformed data. We propose use of a wavelet block thresholding estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.

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Authors and Affiliations

  1. The Wharton School, Department of Statistics, University of Pennsylvania, Philadelphia, PA, 19104-6340, USA

    Lawrence Brown, Tony Cai, Ren Zhang & Linda Zhao

  2. Yale University, New Haven, CT, 06511, USA

    Harrison Zhou

Authors
  1. Lawrence Brown
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  2. Tony Cai
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  3. Ren Zhang
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  4. Linda Zhao
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  5. Harrison Zhou
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Corresponding author

Correspondence to Tony Cai.

Additional information

The research of Tony Cai was supported in part by NSF Grant DMS-0604954.

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Brown, L., Cai, T., Zhang, R. et al. The root–unroot algorithm for density estimation as implemented via wavelet block thresholding. Probab. Theory Relat. Fields 146, 401–433 (2010). https://doi.org/10.1007/s00440-008-0194-2

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  • Received: 23 May 2008

  • Revised: 09 October 2008

  • Published: 06 January 2009

  • Issue Date: March 2010

  • DOI: https://doi.org/10.1007/s00440-008-0194-2

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Keywords

  • Adaptation
  • Block thresholding
  • Coupling inequality
  • Density estimation
  • Nonparametric regression
  • Root–unroot transform
  • Wavelets

Mathematics Subject Classification (2000)

  • Primary: 62G99
  • Secondary: 62F12
  • 62F35
  • 62M99
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