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April 2001 Cross-validation for choosing resolution level for nonlinear wavelet curve estimators
Peter Hall, Spiridon Penev
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Bernoulli 7(2): 317-341 (April 2001).


We show that unless the target density is particularly smooth, cross-validation applied directly to nonlinear wavelet estimators produces an empirical value of primary resolution which fails, by an order of magnitude, to give asymptotic optimality. We note, too, that in the same setting, but for different reasons, cross-validation of the linear component of a wavelet estimator fails to give asymptotic optimality, if the primary resolution level that it suggests is applied to the nonlinear form of the estimator. We propose an alternative technique, based on multiple cross-validation of the linear component. Our method involves dividing the region of interest into a number of subregions, choosing a resolution level by cross-validation of the linear part of the estimator in each subregion, and taking the final empirically chosen level to be the minimum of the subregion values. This approach exploits the relative resistance of wavelet methods to over-smoothing: the final resolution level is too small in some parts of the main region, but that has a relatively minor effect on performance of the final estimator. The fact that we use the same resolution level throughout the region, rather than a different level in each subregion, means that we do not need to splice together different estimates and remove artificial jumps where the subregions abut.


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Peter Hall. Spiridon Penev. "Cross-validation for choosing resolution level for nonlinear wavelet curve estimators." Bernoulli 7 (2) 317 - 341, April 2001.


Published: April 2001
First available in Project Euclid: 25 March 2004

zbMATH: 0981.62031
MathSciNet: MR1828508

Keywords: Curve estimation , Density estimation , generalized kernel methods , Kernel estimator , least-squares cross-validation , linear wavelet estimator , Nonparametric regression , primary resolution level , thresholding

Rights: Copyright © 2001 Bernoulli Society for Mathematical Statistics and Probability


Vol.7 • No. 2 • April 2001
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