Open Access
April 1996 Density estimation by wavelet thresholding
David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, Dominique Picard
Ann. Statist. 24(2): 508-539 (April 1996). DOI: 10.1214/aos/1032894451


Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes $B_{\sigma pq}$ and for a range of global $L'_p$ error measures, $1 \leq p' < \infty$. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when $p' > p$, some form of nonlinearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white-noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error $(p' = 2)$.


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David L. Donoho. Iain M. Johnstone. Gérard Kerkyacharian. Dominique Picard. "Density estimation by wavelet thresholding." Ann. Statist. 24 (2) 508 - 539, April 1996.


Published: April 1996
First available in Project Euclid: 24 September 2002

zbMATH: 0860.62032
MathSciNet: MR1394974
Digital Object Identifier: 10.1214/aos/1032894451

Primary: 62G07 , 62G20

Keywords: adaptive estimation , Besov spaces , Density estimation , minimax estimation , Spatial adaptation , wavelet orthonormal bases

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 2 • April 1996
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