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

  • Lawrence Brown
  • Tony Cai
  • Ren Zhang
  • Linda Zhao
  • Harrison Zhou


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.


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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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Lawrence Brown
    • 1
  • Tony Cai
    • 1
  • Ren Zhang
    • 1
  • Linda Zhao
    • 1
  • Harrison Zhou
    • 2
  1. 1.The Wharton School, Department of StatisticsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Yale UniversityNew HavenUSA

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