Bernoulli

  • Bernoulli
  • Volume 7, Number 2 (2001), 317-341.

Cross-validation for choosing resolution level for nonlinear wavelet curve estimators

Peter Hall and Spiridon Penev

Full-text: Open access

Abstract

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.

Article information

Source
Bernoulli, Volume 7, Number 2 (2001), 317-341.

Dates
First available in Project Euclid: 25 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080222088

Mathematical Reviews number (MathSciNet)
MR1828508

Zentralblatt MATH identifier
0981.62031

Keywords
curve estimation density estimation generalized kernel methods kernel estimator least-squares cross-validation linear wavelet estimator nonparametric regression primary resolution level thresholding

Citation

Hall, Peter; Penev, Spiridon. Cross-validation for choosing resolution level for nonlinear wavelet curve estimators. Bernoulli 7 (2001), no. 2, 317--341. https://projecteuclid.org/euclid.bj/1080222088


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