Predictive density estimators with integrated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e755" altimg="si27.svg"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> loss

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