2022/10/03 by Pankaj Bhagwat, Bhagwat, Pankaj, Éric Marchand +1
Decision Sciences · Mathematics · #62C10 (Secondary) #62C15 #62C20 #62C25 (Primary) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Approximation and Integration #Methodology (stat.ME) #Other Statistics (stat.OT) #Risk and Portfolio Optimization #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2210.00972
openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, q) = ∫ℝd |q(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 qc(y;X) = (1)/(cd) q(‖y-X‖2/c2 ) over the plug-in density q1, 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, q ) ) with strictly increasing γ. The finding is also extended to include scale expansion improvements of more general plug-in densities q(‖y-θ(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 q1 is optimal among all qc.