Minimaxity and Admissibility of Predictive Density Estimators Under S-Hellinger Distances

In this paper, we consider the study of the efficiency of predictive density estimators of multivariate observables measured by the frequentist risk corresponding to S-Hellinger distances as a set of loss functions (for every α ∈ [0, 1]). The main themes, revolve around the inefficiency of minimum risk equivariant (MRE) predictors in high enough dimensions and about the inefficiency of plug-in estimators. We improve the plug-in for a dual point estimation loss with or without expanding the scale. A link between the S-Hellinger distances risk of plug-in type estimators and the risk under reflected normal loss for point estimation is established, bringing into play all the established literature on Stein type dominators. Further, we suggest dominant estimators with or without the presence of restrictions on the unknown mean parameter. Ultimately we prove under the new measure of goodness-of-fit dominance results under a restricted parameter space (multivariate and univariate). Key words : S-Hellinger Distances, Minimaxity, Admissibility, Stein estimation, concave loss, Predictive density estimation