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

The aim of this paper is to discuss a scalar posterior distribution for the shape parameter k of the skew-normal model, when scale and location (or regression) parameters are unknown. The proposed posterior distribution is based on the modified profile likelihood for k (see e.g. Severini, 2000, Chap. 9) and on the corresponding matching prior for k (Ventura et al. 2009). Although the use of a modified profile likelihood or, more generally, of a pseudo-likelihood function for a parameter of interest cannot be considered as orthodox in the Bayesian perspective, it allows to obtain a genuine posterior distribution when combined with a suitable matching prior. Compared to classical Bayesian marginal posterior distribution for k, the proposed approach has the advantage of avoiding elicitation on the nuisance parameters and computation of the integrated likelihood for k. Moreover, the matching prior can be interpreted as a Jeffreys-type prior, since it is proportional to the square root of the profile information. The proposed posterior distribution for k is compared through illustrative examples and simulation studies to other Bayesian analyses on k (see e.g. Liseo and Loperfido, 2006, or Cabras and Castellanos, 2009), showing some interesting results and general conclusions. References - Cabras, S., Castellanos, M.E. (2009), Default Bayesian goodness-of-fit tests for the skew-normal model, J. App. Statist., 32, 223-232. - Liseo, B., Loperfido, N. (2006), Default Bayesian analysis of the skew-normal distribution, J. Statist. Plann. Inf.,136, 373-389. - Severini, T.A. (2000). Likelihood Methods in Statistics, Oxford University Press. - Ventura, L., Cabras, S., Racugno, W. (2009). Prior distributions from pseudo-likelihoods in the presence of nuisance parameters, J. Amer. Statist. Assoc., to appear