Skew-Elliptical Distributions in Bayesian Inference
In this chapter we review recent developments in inference procedures based on skew-elliptical (SE, henceforth) distributions and discuss their impact in Bayesian analysis.
Non-symmetric distributions can be exploited in Bayesian inference both for modeling prior beliefs and for theoretical representation of skewed observables. In the former case, these distributions may represent an honest mathematical formalization of genuine asymmetric information about a vector of parameters of interest; also, they arise in the context of hierarchical linear models when the elicitation process on the location parameters involves linear constraints (O’Hagan and Leonard, 1976; Liseo and Loperﬁdo, 2003a). In the latter case, SE distributions play the same role as they do in classical inference. However, in a frequentist or likelihood setup, inferential procedures for SE classes of densities usually pose challenging problems; as a result, it is not surprising that the literature on skewed distributions has experienced an explosion of papers dealing with distributional properties of these families while devoting minor attention to inferential problems.