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   -> Volume 4, Issue 6


Preprint: Density estimation via wavelets
 
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Brani Vidakovic (brani@isds.Duke.EDU)
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PostPosted: Tue Dec 03, 2002 1:19 pm    
Subject: Preprint: Density estimation via wavelets
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Preprint: Density estimation via wavelets

A paper on density estimation via wavelets is available
in:
pub/brani/papers/
as
SqrtDenWav.ps.

Your comments are most welcome.

Estimating the square root of a density via compactly supported wavelets

Aluisio Pinheiro University of North Carolina -- Chapel Hill
Brani Vidakovic Duke University

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 {v C}encov (1962).
The orthonormal basis used is a basis of compactly supported wavelets
from Daubechies' family. Kerkyacharian and Picard (1992), Donoho et
al. (1993), and Delyon and Juditsky (1993), 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 density to calculate sample
wavelet coefficients. We describe VISUSTOP, a data-driven procedure
for determining the maximum number of levels in the wavelet density
estimator. Coefficients 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.

Key words and phrases:
Wavelets, Density Estimation, Thresholding.

1991 AMS Subject Classification: 62G07

Brani Vidakovic tel. office 919-684-8025 DUKE, ISDS, Box 90251
tel. home 919-309-9638 Durham, NC 27708-0251
All times are GMT + 1 Hour
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