A matching prior for the shape parameter of the skew-normal distribution

This paper deals with the issue of performing a default Bayesian analysis on the shape parameter of the skew-normal distribution. Our approach is based on a suitable pseudo-likelihood function and a matching prior distribution for this parameter, when location (or regression) and scale parameters are unknown. This approach is important for both theoretical and practical reasons. From a theoretical perspective, it is shown that the proposed matching prior is proper thus inducing a proper posterior distribution for the shape parameter, also when the likelihood is monotone. From the practical perspective, the proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals.