Indirect Estimation of alpha-Stable Distributions and Processes

The alpha-stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its practical usefulness is coupled with a marked theoretical appeal, as it stems from a generalized version of the central limit theorem in which the assumption of the finiteness of the variance is replaced by a less restrictive assumption concerning a somehow regular behavior of the tails. Estimation difficulties have however hindered its diffusion among practitioners. Since stably-distributed random numbers can be produced thoroughly, we propose an indirect estimation approach which uses a skew-t distribution as auxiliary model. The properties of this approach are assessed in a detailed simulation study.