Fast cross-validatory choice of wavelet smoothness, primary resolution and threshold in wavelet shrinkage using the Kovac-Silverman algorithm

This article introduces a fast cross-validation algorithm that performs wavelet shrinkage on data sets of arbitrary size and design and also simultaneously selects good values of the primary resolution and number of vanishing moments. We demonstrate the utility of our method by suggesting alternative estimates of the conditional mean of the well-known Ethanol data set. Our alternative estimates outperform the Kovac-Silverman method with a global variance estimate by 25% because of the careful selection of number of vanishing moments and primary resolution. Our alternative estimates are simpler than, and competitive with, results based on the Kovac-Silverman algorithm equipped with a local variance estimate. We include a detailed simulation study that illustrates how our cross-validation method successfully picks good values of the primary resolution and number of vanishing moments for unknown functions based on Walsh functions (to test the response to changing primary resolution) and piecewise polynomials with zero or one derivative (to test the response to function smoothness).