On predictive density estimation with additional information

2017/09/22 by Marchand, Éric, Sadeghkhani, Abdolnasser
#62C20 #62C86 #62F10 #62F15 #62F30 #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1709.07778

Abstract

Based on independently distributed X₁ ∼ N_p(θ₁, σ²₁ I_p) and X₂ ∼ N_p(θ₂, σ²₂ I_p), we consider the efficiency of various predictive density estimators for Y₁ ∼ N_p(θ₁, σ²_Y I_p), with the additional information θ₁ - θ₂ ∈ A and known σ²₁, σ²₂, σ²_Y. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for α-divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss in the univariate case (p=1). An ensemble of techniques are exploited, including variance expansion (for KL loss), point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for q_π_U,A associated with a uniform prior for θ=(θ₁, θ₂) truncated to \θ∈ ℝ^2p: θ₁ - θ₂ ∈ A \, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.

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