MCMC and EM-based methods for inference in heavy-tailed processes with α-stable innovations

In this paper we present both stochastic and deterministic iterative methods for inference about random processes with symmetric stable innovations. The proposed methods use a scale mixtures of normals (SMiN) representation of the symmetric stable law to express the processes in conditionally Gaussian form. This allows standard procedures for dealing with the Gaussian case to be re-used directly as part of the scheme. In contrast with other recently published work on the topic, we propose a novel hybrid rejection sampling method for simulating the scale parameters from their full conditional distributions, making use of asymptotic approximations for the tail of a positive stable distribution when rejection rates are too high. This hybrid approach potentially leads to improved performance compared with straightforward rejection sampling or Metropolis-Hastings (M-H) approaches. The methods can be applied to any model with symmetric stable terms, but we illustrate their application to linear models and present simulations for AR time series with stable innovations.