Indirect estimation of α-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 behaviour 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.