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March 1995 On wavelet methods for estimating smooth functions
Peter Hall, Prakash Patil
Bernoulli 1(1-2): 41-58 (March 1995).

Abstract

Without assuming any prior knowledge of wavelet methods, we develop theory describing their performance as estimators of smooth functions. The linear part of the wavelet estimator is discussed by analogy with classical kernel methods. Concise formulae are developed for its bias, variance and mean square error. These quantities oscillate somewhat erratically on a wavelength that is equivalent to the bandwidth, reflecting the irregular numerical fluctuations that are observed in practice. Nevertheless, the contributions of these oscillations to mean integrated square error tend to dampen one another out, even over very small intervals, with the result that mean integrated square error properties of linear wavelet methods are much closer to those of kernel methods than is perhaps reasonable, given the local behaviour. We illustrate the adaptive qualities of the nonlinear component of a wavelet estimator by describing its performance when the target function is smooth but has high-frequency oscillations. It is shown that the nonlinear component automatically adapts to changing local conditions, to the extent of achieving (except for a logarithmic factor) the same convergence rate as the optimal linear estimator, but without a need to adjust the underlying bandwidth. This makes explicitly clear the way in which the linear part of the estimator takes care of the `average' characteristics of the unknown curve, while the nonlinear part corrects for more erratic fluctuations, in a manner which is virtually independent of the construction of the linear part.

Citation

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Peter Hall. Prakash Patil. "On wavelet methods for estimating smooth functions." Bernoulli 1 (1-2) 41 - 58, March 1995.

Information

Published: March 1995
First available in Project Euclid: 2 August 2007

zbMATH: 0830.62037
MathSciNet: MR1354455

Keywords: convergence rate , Density estimation , differentiability , dilation equation , kernel method , Nonparametric curve estimation , orthogonal series , regression , scaling function , smoothness , ‎wavelet

Rights: Copyright © 1995 Bernoulli Society for Mathematical Statistics and Probability

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Vol.1 • No. 1-2 • March 1995
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