"On Predictive Density Estimation for Location Families under Integrated L₂ andL₁ Losses"

Kubokawa, Tatsuya, Éric Marchand and William E. Strawderman

Our investigation concerns the estimation of predictive densities and a study of effiency as measured by the frequentist risk of such predictive densities with integrated L₂ and L₁ losses. Our findings relate to a p-variate spherically symmetric observable X ∼ p_x (||x -μ||²) and the objective of estimating the density of Y ∼ q_Y (||y - μ||²) based on X. For L₂ loss, we describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three of more dimensions (i.e., p ≥ 3), we show that the MRE estimator is inadmissible under L₂ loss and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of ||δ - μ||². We also study the phenomenon of improvement on the plug-in density estimator of the form q_Y (||y - aX ||²), 0 < a ≤ 1, by a subclass of scale expansions c^-pq_Y (||(y - aX) / c ||²) with c > 1, showing in some cases, inevitably for large enough p, that all choices c > 1 are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for p ≥ 3. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with p ≥ 4 under L₁ loss.