Default Bayesian Analysis of the Skew-Normal Distribution

The Skew Normal (SN, hereafter) class of densities has posed several and interesting inferential problems. In particular, the maximum likelihood estimator of the shape parameter λ may take infinite values with positive sampling probability. To overcome these problems we propose an objective Bayesian approach, based on reference priors. We show that the reference prior for λ is proper when λ is the only parameter in the model; also, the reference prior for λ is marginally proper when location and scale parameters are added. This fact allows us i) to provide a coherent estimation strategy which can be justified also from a pure frequentist viewpoint, since the refrence priors are 2nd order matching priors, and ii) to consider the regular version of the Bayes factor for testing and model selection purposes. In the last years we have experienced a dramatic increase of statistical papers dealing with departures from normality, especially in terms of skewness and tail behaviour. Among the many proposals the SN class of sensities stands for its important mathematical properties, which have facilitated a large number of generalizations of the family in different directions. Despite this fact, many questions remain unsettled, especially from an inferential perspective. The strange behaviour of the maximum likelihood estimator, in fact, is not the only problem: also the method of moments can provide point estimates of the parameters which are outside the admissible range; also, when location and scale parameters are added to the model, the profile likelihood for λ always has a stationary point at λ = 0, independently of the observed data. Our default Bayes approach provides a general solution which overcomes these problems, and it can be easily implemented, as we illustrate with simulated and real data. In particular we build a skew-in-mean GARCH model to represent financial time series data, where innovation terms are assumed to follow a SN distribution. We apply this model to the United Kingdom FTSE index. This paper can be of interest also from a pure theoretical perspective: in fact, it provides the first example of a proper Jeffreys’ (or reference) prior for an unbounded parameter.