Stochastic domination in predictive density estimation for ordered normal means under αα-divergence loss

We consider stochastic domination in predictive density estimation problems when the underlying loss metric is αα-divergence, D(α)D(α), loss introduced by Csiszàr (1967). The underlying distributions considered are normal location-scale models, including the distribution of the observables, the distribution of the variable whose density is to be predicted, and the estimated predictive density which will be taken to be of the plug-in type. The scales may be known or unknown. We derive a general expression for the αα-divergence loss in this set-up and show that it is a concave monotone function of quadratic loss, and also a function of the variances (predicand and plug-in). We demonstrate D(α)D(α) stochastic domination of certain plug-in predictive densities over others for the entire class of metrics simultaneously when üsual” stochastic domination holds in the related problem of estimating the mean with respect to quadratic loss. Examples of D(α)D(α) stochastic domination presented relate to the problem of estimating the predictive density of the variable with the larger mean.