Wavelet Regression By Cross-Validation

This paper is about using wavelets for regression. The main aim of the paper is to introduce and develop a cross-validation method for selecting a wavelet regression threshold that produces good estimates with respect to L2 error. The selected threshold determines which coefficients to keep in an orthogonal wavelet expansion of noisy data and acts in a similar way to a smoothing parameter in non-parametric regression. The paper gives a very brief introduction to wavelets and how the discrete wavelet transform with a thresholding rule can be used to estimate functions from noisy data. The paper examines the integrated square error (ISE) of a soft-thresholded wavelet estimator as a function of the threshold and shows that the ISE is almost always convex and almost always has negative first derivative for a zero threshold. This implies that it is almost always possible to find a unique minimum for the ISE. A cross-validation score is introduced to estimate the ISE of a softthresholded est..