On Bayesian predictive density estimation for skew-normal distributions

This paper is concerned with prediction for skew-normal models, and more specifically the Bayes estimation of a predictive density for Yμ∼SN_p(μ,v_yI_p,λ) under Kullback–Leibler loss, based on Xμ∼SN_p(μ,v_xI_p,λ) with known dependence and skewness parameters. We obtain representations for Bayes predictive densities, including the minimum risk equivariant predictive density p^π_o which is a Bayes predictive density with respect to the noninformative prior π₀≡1. George et al. (Ann Stat 34:78–91, 2006) used the parallel between the problem of point estimation and the problem of estimation of predictive densities to establish a connection between the difference of risks of the two problems. The development of similar connection, allows us to determine sufficient conditions of dominance over p^π_o and of minimaxity. First, we show that p^π_o is a minimax predictive density under KL risk for the skew-normal model. After this, for dimensions p≥3, we obtain classes of Bayesian minimax densities that improve p^π_o under KL loss, for the subclass of skew-normal distributions with small value of skewness parameter. Moreover, for dimensions p≥4, we obtain classes of Bayesian minimax densities that improve p^π_o under KL loss, for the whole class of skew-normal distributions. Examples of proper priors, including generalized student priors, generating Bayesian minimax densities that improve p^π_o under KL loss, were constructed when p≥5. This findings represent an extension of Liang and Barron (IEEE Trans Inf Theory 50(11):2708–2726, 2004), George et al. (Ann Stat 34:78–91, 2006) and Komaki (Biometrika 88(3):859–864, 2001) results to a subclass of asymmetrical distributions.