Robust Wavelet Estimation to Eliminate Simultaneously the Effects of Boundary Problems, Outliers, and Correlated Noise

Classical wavelet thresholding methods suffer from boundary problems caused by the application of the wavelet transformations to a finite signal. As a result, large bias at the edges and artificial wiggles occur when the classical boundary assumptions are not satisfied. Although polynomial wavelet regression and local polynomial wavelet regression effectively reduce the risk of this problem, the estimates from these two methods can be easily affected by the presence of correlated noise and outliers, giving inaccurate estimates. This paper introduces two robust methods in which the effects of boundary problems, outliers, and correlated noise are simultaneously taken into account. The proposed methods combine thresholding estimator with either a local polynomial model or a polynomial model using the generalized least squares method instead of the ordinary one. A primary step that involves removing the outlying observations through a statistical function is considered as well. The practical performance of the proposed methods has been evaluated through simulation experiments and real data examples. The results are strong evidence that the proposed method is extremely effective in terms of correcting the boundary bias and eliminating the effects of outliers and correlated noise. 1. Introduction Suppose a noisy data set lives on the fixed design model The classical model (1.1) assumes that the unknown function, , is square integrable on the interval . The sequence is independent and distributed normally with identical means zero and identical variances . Wavelet methods have been intensively used over the last two decades for estimating an unknown function observed in the presence of noise, following the pioneering work presented in the seminal papers of Donoho and Johnstone [1, 2], where the concept of wavelet thresholding was introduced to the statistical literature. Researchers have begun looking at the situation where the usual assumptions are no longer satisfied. When the noise contains a certain amount of structure in the form of a correlation, the variances of the wavelet coefficients will depend on the resolution level of the wavelet decomposition but will be constant at each level. As a result, the global thresholding breaks down because of its great difficulty in providing a global threshold value to threshold wavelet coefficients at all the desired levels. In order to overcome this limitation, Johnstone and Silverman [3] recommended using level by level thresholding, that is, not to use only one thresholding value for all coefficients at all