Estimating the Square Root of a Density Via

A large body of nonparametric statistical literature is devoted to density estimation. Overviews are given in Silverman (1986) and Izenman (1991). This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by Cencov (1962). The orthonor-mal basis used is a basis of compactly supported wavelets from Daubechies' family. among others, applied wavelets in density estimation. The local nature of wavelet functions makes the wavelet estimator superior to projection estimators that use classical orthonormal bases (Fourier, Hermite, etc.) Instead of estimating the unknown density directly, we estimate the square root of the density, which enables us to control the positiveness and the L 1-norm of the density estimate. However, in that approach one needs a pre-estimator of the density to calculate sample wavelet coeecients. We describe VISUSTOP, a data-driven procedure for determining the maximum number of levels in the wavelet density estimator. Coeecients in the selected levels are thresholded to make the estimator parsimonious. Our method is illustrated on the Galaxy velocity data set (Roeder, 1990) and implemented in S-Plus. The method can be readily extended to a multidimensional case and other wavelet bases.