Regularization of Wavelet Approximations
Top Cited Papers
- 1 September 2001
- journal article
- Published by Taylor & Francis in Journal of the American Statistical Association
- Vol. 96 (455) , 939-967
- https://doi.org/10.1198/016214501753208942
Abstract
In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are specific members of nonlinear regularized wavelet estimators. They correspond to the lower and upper envelopes of a class of the penalized least squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax. To efficiently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRSI) are proposed as initial estimators, wh...Keywords
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