On predictive density estimation with additional information

Based on independently distributed $X_1\sim N_p({\theta }_1,{\sigma }_1^2{I}_p)$ and $X_2\sim N_p({\theta }_2,{\sigma }_2^2{I}_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(\theta_1, \sigma^2_Y I_p)$,withtheadditionalinformation$\theta_1 - \theta_2 \in A$andknown$\sigma^2_1, \sigma^2_2, \sigma^2_Y$. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for $\alpha -$divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss in the univariate case ($p=1$). An ensemble of techniques are exploited, including variance expansion (for KL loss), point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for ${\widehat{q}}_{{\pi }_{U,A}}$ associated with a uniform prior for $\theta=(\theta_1, \theta_2)$truncatedto${\theta \in $\mathbb{R}$^{2p}: \theta_1 - \theta_2 \in A }$, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.