Title: On predictive density estimation with additional information,
Abstract: Based on independently distributed $X_1 \sim N_p(\theta_1, \sigma^2_1 I_p)$
and $X_2 \sim N_p(\theta_2, \sigma^2_2 I_p)$, we consider the efficiency of
various predictive density estimators for $Y_1 \sim N_p(\theta_1, \sigma^2_Y
I_p)$, with the additional information $\theta_1 - \theta_2 \in A$ and known
$\sigma^2_1, \sigma^2_2, \sigma^2_Y$. 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
$\hat{q}_{\pi_{U,A}}$ associated with a uniform prior for $\theta=(\theta_1,
\theta_2)$ truncated to $\{\theta \in \mathbb{R}^{2p}: \theta_1 - \theta_2 \in
A \}$, 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.