Date of this Version
Probability Theory and Related Fields
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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-008-0194-2.
adaptation, block thresholding, coupling inequality, density estimation, nonparametric regression, root-unroot transform, wavelets
Brown, L., Cai, T., Zhang, R., Zhao, L., & Zhou, H. (2010). The Root–Unroot Algorithm for Density Estimation as Implemented via Wavelet Block Thresholding. Probability Theory and Related Fields, 146 401-433. http://dx.doi.org/10.1007/s00440-008-0194-2
Date Posted: 27 November 2017
This document has been peer reviewed.