Predictive density estimators with integrated L_1 loss
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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∈ℝ^d. The chosen criteria are integrated L_1 loss given by L(θ, q̂) = ∫_ℝ^d|q̂(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 q̂_c(y;X) = 1/c^d q(y-X^2/c^2 ) over the plug-in density q̂_1, for a subset of values c ∈ (1,c_0). The finding is universal with respect to p,q, and d ≥ 2, and extended to loss functions γ(L(θ, q̂ ) ) with strictly increasing γ. The finding is also extended to include scale expansion improvements of more general plug-in densities q(y-θ̂(X)^2 ), when the parameter space Θ is a compact subset of ℝ^d. 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 q̂_1 is optimal among all q̂_c.
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