On efficient prediction and predictive density estimation for spherically symmetric models
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Let X,U,Y be spherically symmetric distributed having density η^d +k/2 f(η(x-θ|^2+ u^2 + y-cθ^2 ) ) , with unknown parameters θ∈R^d and η>0, and with known density f and constant c >0. Based on observing X=x,U=u, we consider the problem of obtaining a predictive density q̂(y;x,u) for Y as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant density q̂_mre, which is Generalized Bayes with respect to the prior π(θ, η) = η^-1. For d ≥ 3, we obtain improvements on q̂_mre, and further show that the dominance holds simultaneously for all f subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior π_h(θ, η) =η^-1θ^2-d dominates q̂_mre simultaneously for all scale mixture of normals f.
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