SCHOOL OF STATISTICS

BUEHLER-MARTIN DISTINGUISHED LECTURER SERIES

Established by Mr. and Mrs. Thomas Martin

In memory of Robert J. Buehler, Professor of Statistics 1963-1988

A Root-Unroot Algorithm for Nonparametric Density Estimation

and an Implementation via Adaptive Wavelet Block Threshholding

Lawrence D. Brown

Statistics Department, Wharton School

University of Pennsylvania

Thursday, May 1, 2008

3:30 – 4:30 pm Ford 155

Refreshments 3:00 Ford 300

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. The resulting density estimator enjoys good finite sample performance and a high degree of asymptotic adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this density estimation procedure.

Proof of asymptotic adaptivity is more challenging than implementation of the procedure. The proof will only be sketched. The first step is a Poissonization argument to show that the fixed sample size density problem is not essentially different from the density problem where the sample size is a Poisson random variable. The second step is to extend the quantile coupling inequality of Komlos, Major and Tusnady (1975) to approximate the binned and root transformed data by independent normal variables. The third step is the derivation of a risk bound for block thresholding in the case where the noise is not necessarily Gaussian. A further argument is needed to justify the final normalization step in the algorithm.

This is joint work with T. T. Cai, R. Zhang, L. Zhao, and H. Zhou