On wavelet methods for estimating smooth functions

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.