Nonparametric Wavelet Thresholding and Profile Monitoring for Non-Gaussian Errors

Recent advancements in data collection allow scientists and researchers to obtain massive amounts of information in short periods of time. Often this data is functional and quite complex. Wavelet transforms are popular, particularly in the engineering and manufacturing fields, for handling these type of complicated signals. A common application of wavelets is in statistical process control (SPC), in which one tries to determine as quickly as possible if and when a sequence of profiles has gone out-of-control. However, few wavelet methods have been proposed that don't rely in some capacity on the assumption that the observational errors are normally distributed. This dissertation aims to fill this void by proposing a simple, nonparametric, distribution-free method of monitoring profiles and estimating changepoints. Using only the magnitudes and location maps of thresholded wavelet coefficients, our method uses the spatial adaptivity property of wavelets to accurately detect profile changes when the signal is obscured with a variety of non-Gaussian errors. Wavelets are also widely used for the purpose of dimension reduction. Applying a thresholding rule to a set of wavelet coefficients results in a "denoised" version of the original function. Once again, existing thresholding procedures generally assume independent, identically distributed normal errors. Thus, the second main focus of this dissertation is a nonparametric method of thresholding that does not assume Gaussian errors, or even that the form of the error distribution is known. We improve upon an existing even-odd cross-validation method by employing block thresholding and level dependence, and show that the proposed method works well on both skewed and heavy-tailed distributions. Such thresholding techniques are essential to the SPC procedure developed above.