Predictive density estimators with integrated L1 loss

This paper addresses the problem of an efficient predictive density estimation for the density q(‖y−θ‖2) of Y based on X∼p(‖x−θ‖2) for y,x,θ∈Rd. The chosen criteria are integrated L1 loss given by L(θ,qˆ)=∫Rd|qˆ(y)−q(‖y−θ‖2)|dy, and the associated frequentist risk, for θ∈Θ. For absolutely continuous and strictly decreasing q, we establish the inevitability of scale expansion improvements qˆc(y;X)=1cdq(‖y−X‖2/c2) over the plug-in density qˆ1, for a subset of values c∈(1,c0). The finding is universal with respect to p,q, and d≥2, and extended to loss functions γ(L(θ,qˆ)) with strictly increasing γ. The finding is also extended to include scale expansion improvements of more general plug-in densities q(‖y−θˆ(X)‖2), when the parameter space Θ is a compact subset of Rd. Numerical analyses illustrative of the dominance findings are presented and commented upon. As a complement, we demonstrate that the unimodal assumption on q is necessary with a detailed analysis of cases where the distribution of Y|θ is uniformly distributed on a ball centered about θ. In such cases, we provide a univariate (d=1) example where the best equivariant estimator is a plug-in estimator, and we obtain cases (for d=1,3) where the plug-in density qˆ1 is optimal among all qˆc.