Statistics Papers

Document Type

Journal Article

Date of this Version

3-2010

Publication Source

Probability Theory and Related Fields

Volume

146

Start Page

401

Last Page

433

DOI

10.1007/s00440-008-0194-2

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.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-008-0194-2.

Keywords

adaptation, block thresholding, coupling inequality, density estimation, nonparametric regression, root-unroot transform, wavelets

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Date Posted: 27 November 2017

This document has been peer reviewed.