Abstract
Here we present theoretical results on the random wavelet coefficients covariance structure. We use simple properties of the coefficients to derive a recursive way to compute the within- and across-level covari-ances. We then show the usefulness of those findings in some of the best known applications of wavelets in statistics. Wavelet shrinkage attempts to estimate a function from noisy data. When approaching the problem from a Bayesian point of view, a prior distribution is imposed on the coefficients of the unknown function. We show how it is possible to specify priors that take into account the full correlation among coefficients through a parsimonious number of hyperparameters. We then concentrate on the wavelet analysis of random processes. Given discrete measurements from a long-memory process, a wavelet transform leads to a set of wavelet coefficients. We show how the statistical properties of the coefficients depend on the process underlying the data and how a Bayesian approach gives rise to a natural way of including in the model information about the process itself.
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Vannucci, M., Corradi, F. (1999). Modeling Dependence in the Wavelet Domain. In: Müller, P., Vidakovic, B. (eds) Bayesian Inference in Wavelet-Based Models. Lecture Notes in Statistics, vol 141. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0567-8_12
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DOI: https://doi.org/10.1007/978-1-4612-0567-8_12
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