Penalized wavelets: Embedding wavelets into semiparametric regression

Marginal longitudinal semiparametric regression via penalized splines.

We study the marginal longitudinal nonparametric regression problem and some of its semiparametric extensions. We point out that, while several elaborate proposals for efficient estimation have been proposed, a relative simple and straightforward one, based on penalized splines, has not. After describing our approach, we then explain how Gibbs sampling and the BUGS software can be used to achie...

On Semiparametric Regression with O'sullivan Penalized Splines

An exposition on the use of O’Sullivan penalized splines in contemporary semiparametric regression, including mixed model and Bayesian formulations, is presented. O’Sullivan penalized splines are similar to P-splines, but have the advantage of being a direct generalization of smoothing splines. Exact expressions for the O’Sullivan penalty matrix are obtained. Comparisons between the two types o...

Real nonparametric regression using complex wavelets

Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform (DWT) of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; and then invert the DWT to form an estimate of the...

Wavelets for Regression and Other Statistical Problems

Regression in random design and warped wavelets

We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis {ψjk(G), j, k} warped with the design. This allows to perform a very stable and computable thresholding algorithm. We investigate the properties of this new basis. In particular, we prove that if the desig...