Wavelet Shrinkage for Regression Models with Uniform Design and Correlated Errors

This paper presents some results on semi-parametric regression using wavelet methods in the presence of autocorrelated stationary Gaussian errors, and when the explanatory variable follows an Uniform distribution. It is shown that this Uniform distribution arrives in the special cases of stochastic sampling: the Poisson and the jittered sampling schemes. The aim is to estimate the signal globally with low risk. It is shown that in these special cases the samples can be treated as if they were equispaced and with correlated noise; i.e., the estimator achieves an almost optimal convergence rate. Some simulation studies compare the cases with and without correlated errors in finite samples. Daily volatility estimatives of a low-traded stock illustrate the usefulness of the method.