The Root-Unroot Algorithm for Density Estimation as Implemented via Wavelet Block Thresholding

Lawrence Brown, Tony Cai, Ren Zhang, Linda Zhao, and Harrison Zhou

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• Abstract: Density estimation has traditionally been treated separately from nonparametric regression. In this paper we propose and implement a density estimation procedure which begins by turning density estimation into a 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. A nonparametric regression estimator is then applied to the transformed data. Finally, the estimated regression function is un-rooted by squaring and normalizing.

  In principle many common nonparametric regression estimators could be used in the implementation of this algorithm. We propose use of a wavelet block thresholding estimator. The entire algorithm is then convenient to implement. We show that the resulting density estimator enjoys a high degree of adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this density estimation procedure.

• Paper: pdf file.

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