Adaptation to high spatial inhomogeneity using wavelet methods

Many of the signals to which wavelet methods are applied, including those encountered in simulation experiments, are essentially smooth but contain a small number of high-frequency episodes such as spikes. In principle it is possible to employ a different amount of smoothing at different spatial locations, but in the context of wavelets this is so awkward to implement that it is not really practicable. Instead, it is attractive to select the primary resolution level (or smoothing parameter) so as to give good performance for smooth parts of the signal. While this is readily accomplished using a cross-validation argument, it is unclear whether it has a deleterious impact on performance at high-frequency episodes. In this paper we show that it does not. We derive upper and lower bounds to pointwise rates of convergence for functions whose "spikiness" increases with sample size. (This allows us to model contexts where wavelet methods have to work hard to recover high-frequency events.) We show that, in order to achieve optimal rates of convergence, it is necessary for the primary resolution level of the empirical wavelet transform to vary with location, sometimes extensively. Nevertheless, the convergence rate penalty incurred through using a non-varying resolution level, chosen to provide good performance for coarse-scale features, equals a factor that is less than the logarithm of sample size.