Inference in symmetric alpha-stable noise using MCMC and the slice sampler

We have previously shown how to perform inference about symmetric stable processes using Monte Carlo EM (MCEM) and Markov chain Monte Carlo (MCMC) techniques. Simulation based methods such as these are an excellent tool for inference with stable law distributions, since they do not require any direct evaluation of the stable density function, which is unavailable analytically in the general case. We review the existing methods for inference with MCMC and propose new methods based on the slice sampler, a very simple sampling algorithm which draws points from a uniform distribution over the area under the required density function. There is some evidence in the literature that the slice sampler has better convergence properties than the independence Metropolis samplers and rejection samplers previously proposed. We investigate this in the context of alpha-stable noise distributions.