  In this work, we are concerned with the estimation of the predictive density
of a Gaussian random vector where both the mean and the variance are unknown.
In such a context, we prove the inadmissibility of the best equivariant
predictive density under the Kullback-Leibler risk in a nonasymptotic
framework. Our result stands whatever the dimension d of the vector is, even
when d<=2, which can be somewhat surprising compared to the known variance
setting. We also propose a class of priors leading to a Bayesian predictive
density that dominates the best equivariant one. Throughout the article, we
give several elements that we believe are useful for establishing the parallel
between the prediction and the estimation problems, as it was done in the known
variance framework.
