Bayesian Inference for Skewed Stable Distributions

Stable distributions are a class of distributions which allow skewness and heavy tail. Non-Gaussian stable random variables play the role of normal distribution in the central limit theorem, for normalized sums of random variables with infinite variance. The lack of analytic formula for density and distribution functions of stable random variables has been a major drawback to the use of stable distributions, also in the case of inference in Bayesian framework. Buckle introduced priors for the parameters of stable random variables to obtain an analytic form of posterior distribution. However, many researchers tried to solve the problem, through the Markov chain Monte Carlo methods, e.g. [8] and their references. In this paper a new class of heavy-tailed distribution is introduced, called skewed stable. This class has two main advantages: It has many inferential advantages, since it is a member of exponential family, so the Bayesian inference can be drawn similar to the exponential family of distributions and modelling skew data with stable distributions is dominated by this family. Finally, Bayesian inference for skewed stable are compared to the stable distributions through a few simulations study.