On Bayesian predictive density estimation for skew-normal distributions

Abstract

This paper is concerned with prediction for skew-normal models, and more specifically the Bayes estimation of a predictive density for \(Y \left. \right| \mu \sim {\mathcal {S}} {\mathcal {N}}_p (\mu , v_y I_p, \lambda )\) under Kullback–Leibler loss, based on \(X \left. \right| \mu \sim {\mathcal {S}} {\mathcal {N}}_p (\mu , v_x I_p, \lambda )\) with known dependence and skewness parameters. We obtain representations for Bayes predictive densities, including the minimum risk equivariant predictive density \(\hat{p}_{\pi _{o}}\) which is a Bayes predictive density with respect to the noninformative prior \(\pi _0\equiv 1\). 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 \(\hat{p}_{\pi _{o}}\) and of minimaxity. First, we show that \(\hat{p}_{\pi _{o}}\) is a minimax predictive density under KL risk for the skew-normal model. After this, for dimensions \(p\ge 3\), we obtain classes of Bayesian minimax densities that improve \(\hat{p}_{\pi _{o}}\) under KL loss, for the subclass of skew-normal distributions with small value of skewness parameter. Moreover, for dimensions \(p\ge 4\), we obtain classes of Bayesian minimax densities that improve \(\hat{p}_{\pi _{o}}\) 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 \(\hat{p}_{\pi _{o}}\) under KL loss, were constructed when \(p\ge 5\). 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.