Wavelet Shrinkage for Regression Models with Random Design and Correlated Errors

Extraction of a signal in the presence of sto\-chastic noise via wavelet shrinkage has been studied under different assumptions about both the statistical properties of the noise and the pattern of the locations at which the noisy signal is observed. The simplest assumptions are that the noise is independent and identically distributed (IID) and that the samples are equispaced (evenly spaced in time). Previous work has relaxed either the IID assumption to allow for correlated observations or the equispaced assumption to allow for random sampling, but very few papers has relaxed both together. In this paper we relax both assumptions by assuming the noise to be a stationary Gaussian process (with mild restrictions on its autocorrelation sequence) 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 (i.e., the random sampling is simply ignored), the resulting wavelet-based shrinkage estimator achieves an almost optimal convergence rate. We investigate the efficacy of the proposed methodology via simulation studies and extraction of the light curve for a variable star.