A note on Bayesian hypothesis testing for the scalar skew-normal distribution

The skew-normal distribution is a class of densities that preserves some useful properties of the normal distribution while allowing a shape parameter to account for skewness. It has various remarkable properties in terms of mathematical tractability and turned out to be quite useful in modelling real data. However from an inferential point of view its use gives raise to many difficulties, that are intrinsically tied with the shape of the likelihood function. This fact suggests to solve the problem by calibrating the likelihood with a weight function, and perhaps the most intuitive calibration can be obtained in the Bayesian framework, where the prior distribution plays naturally the role of the weight function. Here we consider in details the problem of testing normality in the general skew-normal model, and solve it by means of different tools for hypothesis testing in the Bayesian framework, namely the Bayes factor and the Jeffreys divergence, pointing out benefits and problems of both approaches