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

Inadmissibility of the best equivariant predictive density in the unknown variance case