Wavelet Packet Thresholding and Spectrum Estimation

We consider the recent suggestion that spectrum estimation can be accomplished by applying wavelet denoising methodology to wavelet packet coefficients derived from the logarithm of a spectrum estimate. The particular algorithm we consider consists of computing the logarithm of the multitaper spectrum estimator,applying an orthonormal transform derived from a wavelet packet table to the log multitaper spectrum ordinates, thresholding the empirical wavelet packet coefficients,and then inverting the transform. For a small number of tapers suitable partitions/bases for different stationary time series are all similar,and easily derived,and any differences between the wavelet packet and DWT approaches are minimal. For a larger number of tapers, where the chosen parameters satisfy the conditions of a proven theorem,nothing can be gained over the simpler discrete wavelet transform (DWT) thresholding approach. We thus conclude that the DWT approach is a very adequate wavelet-based approach,and that nothing substantial will be gained by using more complicated wavelet packets

On some stationary models: construction and estimation

We propose a simple yet powerful method to construct strictly stationary Markovian models with given, but arbitrary, invariant distributions. The idea is based on a Poisson transform modulating the dependence structure in the model. An appealing feature of our approach is that we are able to fully control the underlying transition probabilities and therefore incorporate them within standard estimation methods. We analyze some specific cases in both discrete and continuous time. Given our proposed representation of the transition density, a Gibbs sampler algorithm, based on the slice method, is proposed and implemented. In particular, the resulting methodology is of interest for the estimation of certain continuous time models, such as diffusion processes