A note on the reference prior for the scalar skew-normal distribution

In this note, we discuss some peculiar features of default Bayes analysis of the scalar skew-normal model. In particular we show that, when we consider the simplest model with a single skewness parameter lambda unknown, the reference —or Jeffreys’—prior for this parameter, whose support is unbounded, is proper. From a Bayesian perspective this is important because the likelihood function for lambda happens to be monotone with positive sampling probability and the use of improper priors would be impossible. We also explore the frequentist properties of the default Bayes approach. Finally, we consider the general scalar case with unknown location and scale parameters and we derive a closed form expression for the integrated likelihood function for when the location and scale parameters are integrated out with respect to the usual conditional reference prior.