We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter , and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for θ. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on θ, and illustrate and comment the findings based on numerical evaluations.
Éric Marchand’s research is supported by the Natural Sciences and Engineering Research Council of Canada. Pankaj Bhagwat is grateful to the ISM (Institut des sciences mathématiques) for financial support.
Thanks to Jean-Philippe Burelle for useful discussions on geometric representations related to prior density (4.9). Finally, thanks to two reviewers, an Associate Editor, and Editor Grace Yi for valuable comments and suggestions throughout the evaluation process.
"Bayesian inference and prediction for mean-mixtures of normal distributions." Electron. J. Statist. 17 (2) 1893 - 1922, 2023. https://doi.org/10.1214/23-EJS2142