Wavelet shrinkage for regression models with random design and correlated errors

Abstract

Extraction of a signal in the presence of stochastic noise via wavelet shrinkage has been studied under assumptions that the noise is independent and identically distributed (IID) and that the samples are equispaced (evenly spaced in time). Previous work has relaxed these assumptions either to allow for correlated observations or to allow for random sampling, but very few papers have relaxed both together. In this paper we relax both assumptions by assuming the noise to be a stationary Gaussian process and by assuming a random sampling scheme dictated either by a uniform distribution or by an evenly spaced design subject to jittering. We show that, if the data are treated as if they were autocorrelated and equispaced, the resulting wavelet-based shrinkage estimator achieves an almost optimal convergence rate. We investigate the efficacy of the proposed methodology via simulation studies and illustrate it by the extraction of the light curve for a variable star.